diff options
Diffstat (limited to 'backend')
41 files changed, 14 insertions, 23329 deletions
diff --git a/backend/Affinity_relation.v b/backend/Affinity_relation.v deleted file mode 100755 index 7c4029c..0000000 --- a/backend/Affinity_relation.v +++ /dev/null @@ -1,152 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Delete_Preference_Edges_Degree. -Require Import Edges. -Require Import MyRegisters. - -Module Register := Regs. - -Import Edge RegFacts Props. - -Lemma compat_bool_move : forall g, -compat_bool Register.eq (move_related g). - -Proof. -unfold move_related, compat_bool. intros. -rewrite (compat_preference_adj _ _ _ H). reflexivity. -Qed. - -(* Characterization of the move-relationship *) - -(* If a vertex x is move-related in g then there exists an affinity edge e - of g which is incident to x *) -Lemma move_related_charac : forall x g, -move_related g x = true -> -exists e, aff_edge e /\ In_graph_edge e g /\ incident e x. - -Proof. -intros. unfold move_related in H. -destruct (set_induction2 (preference_adj x g)). -rewrite Dec.F.is_empty_iff in H0. rewrite H0 in H. inversion H. -destruct H0. destruct H0. -rewrite Add_Equal in H0. -assert (VertexSet.In x0 (preference_adj x g)). -rewrite H0. apply VertexSet.add_1. intuition. -rewrite in_pref in H1. destruct H1. -exists (x0, x, Some x2). -split. unfold aff_edge. exists x2. auto. -split. assumption. -right. auto. -Qed. - -(* The inversion characterization of move related *) -Lemma move_related_charac2 : forall x g e, -aff_edge e -> -In_graph_edge e g -> -incident e x -> -move_related g x = true. - -Proof. -intros. unfold move_related. -case_eq (VertexSet.is_empty (preference_adj x g)); auto. -intros. rewrite <-Dec.F.is_empty_iff in H2. -generalize (empty_is_empty_1 H2). clear H2. intro. -destruct H. destruct H1. -assert (VertexSet.In (snd_ext e) (preference_adj x g)). -rewrite in_pref. exists x0. rewrite edge_comm. rewrite <-H. -assert (eq (x, snd_ext e, get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_eq. -rewrite H3. rewrite <-(edge_eq e). assumption. -rewrite H2 in H3. elim (VertexSet.empty_1 H3). -assert (VertexSet.In (fst_ext e) (preference_adj x g)). -rewrite in_pref. exists x0. rewrite <-H. -assert (eq (fst_ext e, x, get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_eq. -rewrite H3. rewrite <-(edge_eq e). assumption. -rewrite H2 in H3. elim (VertexSet.empty_1 H3). -Qed. - -(* Characterization of nonmove relation *) -Lemma move_related_false_charac : forall e x g, -aff_edge e -> -In_graph_edge e g -> -move_related g x = false -> -~incident e x. - -Proof. -intros e x g H H0 H1 H2. -generalize (move_related_charac2 _ _ _ H H0 H2). -intro H3;rewrite H1 in H3;inversion H3. -Qed. - -Lemma move_related_false_charac2 : forall x g, -(forall e, aff_edge e -> In_graph_edge e g -> ~incident e x) -> -move_related g x = false. - -Proof. -intros x g H. -case_eq (move_related g x);intro H0. -generalize (move_related_charac _ _ H0);intro H1. -destruct H1 as [e H1];destruct H1 as [H1 H2];destruct H2 as [H2 H3]. -elim ((H e H1 H2) H3). -reflexivity. -Qed. - -(* A move-related vertex of the graph belongs to the graph *) -Lemma move_related_in_graph : forall x g, -move_related g x = true -> In_graph x g. - -Proof. -intros x g H. -generalize (move_related_charac x g H);intro H0. -destruct H0 as [e H0];destruct H0 as [H0 H1];destruct H1 as [H1 H2]. -destruct H2;rewrite H2. -apply (proj1 (In_graph_edge_in_ext _ _ H1)). -apply (proj2 (In_graph_edge_in_ext _ _ H1)). -Qed. - -(* The endpoints of any affinity edge of g are move-related in g *) -Lemma Aff_edge_aff : forall e g, -In_graph_edge e g -> -aff_edge e -> -move_related g (fst_ext e) = true /\ move_related g (snd_ext e) = true. - -Proof. -intros. split. -apply move_related_charac2 with (e:=e); [idtac|idtac|left]; auto. -apply move_related_charac2 with (e:=e); [idtac|idtac|right]; auto. -Qed. - -(* Any move-related vertex has a nonempty preference neighborhood *) -Lemma move_related_not_empty_pref : forall x g, -move_related g x = true -> -~VertexSet.Equal (preference_adj x g) VertexSet.empty. - -Proof. -unfold move_related. intros. intro. -generalize (empty_is_empty_2 H0). intro. -rewrite Dec.F.is_empty_iff in H1. rewrite H1 in H. inversion H. -Qed. - -(* Any nonmove-related vertex has an empty preference neighborhood *) -Lemma not_move_related_empty_pref : forall x g, -move_related g x = false -> -VertexSet.Equal (preference_adj x g) VertexSet.empty. - -Proof. -unfold move_related. intros. -apply empty_is_empty_1. rewrite Dec.F.is_empty_iff. -case_eq (VertexSet.is_empty (preference_adj x g)); intros; auto. -rewrite H0 in H. inversion H. -Qed. - -(* Any vertex having a preference degree different than 0 is move-related *) -Lemma move_related_card : forall x g, -pref_degree g x <> 0 -> -move_related g x = true. - -Proof. -unfold pref_degree. intros. -case_eq (move_related g x); intros. -reflexivity. -generalize (not_move_related_empty_pref _ _ H0). intro. -rewrite H1 in H. rewrite empty_cardinal in H. elim H. auto. -Qed. diff --git a/backend/Allocproof.v b/backend/Allocproof.v index c01b80c..10eaa5b 100644 --- a/backend/Allocproof.v +++ b/backend/Allocproof.v @@ -711,9 +711,7 @@ Proof. simpl. econstructor; eauto. change (transfer f (RTL.fn_entrypoint f) live !! (RTL.fn_entrypoint f)) with (live0 f live). - eapply agree_parameters; eauto. - unfold regalloc. eauto. - unfold regalloc. congruence. + eapply agree_parameters; eauto. (* external function *) injection H7; intro EQ; inv EQ. diff --git a/backend/Coloring.v b/backend/Coloring.v index 5282d48..67824ae 100644 --- a/backend/Coloring.v +++ b/backend/Coloring.v @@ -22,8 +22,6 @@ Require Import RTLtyping. Require Import Locations. Require Import Conventions. Require Import InterfGraph. -Require Import MyAllocation. -Require Import InterfGraph_Construction. (** * Construction of the interference graph *) @@ -92,7 +90,7 @@ Require Import InterfGraph_Construction. - between the result of a ``call'' instruction and the location of the result as dictated by the calling conventions. *) -(* + Definition add_interf_live (filter: reg -> bool) (res: reg) (live: Regset.t) (g: graph): graph := Regset.fold @@ -211,10 +209,10 @@ Definition interf_graph (f: function) (live: PMap.t Regset.t) (live0: Regset.t) and the set of all [RTL] pseudo-registers mentioned in the interference graph. It returns the coloring as a function from pseudo-registers to locations. *) -(* + Parameter graph_coloring: function -> graph -> regenv -> Regset.t -> (reg -> loc). -*) + (** To ensure that the result of [graph_coloring] is a correct coloring, we check a posteriori its result using the following Coq functions. Let [coloring] be the function [reg -> loc] returned by [graph_coloring]. @@ -268,7 +266,7 @@ Definition check_coloring andb (check_coloring_1 g coloring) (andb (check_coloring_2 g coloring) (check_coloring_3 rs env coloring)). -*) + (** To preserve decidability of checking, the checks (especially the third one) are performed for the pseudo-registers mentioned in the interference graph. To facilitate the proofs, @@ -292,14 +290,11 @@ Definition alloc_of_coloring (coloring: reg -> loc) (env: regenv) (rs: Regset.t) and adjustment of this coloring. If the coloring candidate is incorrect, [None] is returned, causing register allocation to fail. *) -Definition graph_coloring (f : function) g env (rs : Regset.t) := -my_graph_coloring g env. - Definition regalloc (f: function) (live: PMap.t Regset.t) (live0: Regset.t) (env: regenv) := let g := interf_graph f live live0 in let rs := all_interf_regs g in let coloring := graph_coloring f g env rs in -(* if check_coloring g env rs coloring *) -(* then *) Some (alloc_of_coloring coloring env rs). -(* else None. *) + if check_coloring g env rs coloring + then Some (alloc_of_coloring coloring env rs) + else None. diff --git a/backend/Coloringproof.v b/backend/Coloringproof.v index cc4234e..92ac067 100644 --- a/backend/Coloringproof.v +++ b/backend/Coloringproof.v @@ -24,8 +24,6 @@ Require Import Locations. Require Import Conventions. Require Import InterfGraph. Require Import Coloring. -Require Import InterfGraph_Construction. -Require Import MyAllocation. (** * Correctness of the interference graph *) @@ -448,12 +446,12 @@ Proof. intros until g2. intro. unfold correct_interf_instr; destruct instr; auto. destruct (is_move_operation o l). - intros. apply interfere_incl with (g1 := g1); auto. - intros. apply interfere_incl with (g1 := g1); auto. - intros. apply interfere_incl with (g1 := g1); auto. + intros. eapply interfere_incl; eauto. + intros. eapply interfere_incl; eauto. + intros. eapply interfere_incl; eauto. intros [A [B C]]. - split. intros. apply interfere_mreg_incl with (g1 := g1); auto. - split. intros. apply interfere_incl with (g1 := g1); auto. + split. intros. eapply interfere_mreg_incl; eauto. + split. intros. eapply interfere_incl; eauto. destruct s0; auto. intros. eapply interfere_mreg_incl; eauto. destruct s0; auto. intros. eapply interfere_mreg_incl; eauto. Qed. @@ -814,23 +812,16 @@ Let g := interf_graph f live live0. Let allregs := all_interf_regs g. Let coloring := graph_coloring f g env allregs. - Lemma regalloc_ok: regalloc f live live0 env = Some alloc -> check_coloring g env allregs coloring = true /\ alloc = alloc_of_coloring coloring env allregs. Proof. -unfold regalloc. intro. -inversion H. subst. clear H. -split. apply allocation_correct. -auto. -(* unfold regalloc, coloring, allregs, g. case (check_coloring (interf_graph f live live0) env). intro EQ; injection EQ; intro; clear EQ. split. auto. auto. intro; discriminate. -*) Qed. Lemma regalloc_acceptable: diff --git a/backend/Conservative_criteria.v b/backend/Conservative_criteria.v deleted file mode 100755 index 946a047..0000000 --- a/backend/Conservative_criteria.v +++ /dev/null @@ -1,339 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Affinity_relation. -Require Import ZArith. -Require Import Edges. -Require Import MyRegisters. - -Import Edge RegFacts Props. - -(* Definition of the Briggs criterion *) - -Definition nb_of_significant_degree g K s := -VertexSet.fold (fun x n => if (le_lt_dec K (interf_degree g x)) then S n else n) - s 0. - -(* Definition of the conservative criteria : an affinity edge e is coalescible if - 1) one of its endpoints is not precolored - 2) the endpoints have less than K interference neighbors of high-degree *) -Definition conservative_coalescing_criteria e g K := -if (is_precolored (fst_ext e) g && is_precolored (snd_ext e) g) then false else -if le_lt_dec -K -(nb_of_significant_degree g K - (VertexSet.union (interference_adj (fst_ext e) g) - (interference_adj (snd_ext e) g))) -then false else true. - -(* Specification of the coalescing criterion *) -Lemma conservative_coalescing_criteria_1 : forall e g K, -~VertexSet.In (fst_ext e) (precolored g) \/ -~VertexSet.In (snd_ext e) (precolored g) -> -In_graph_edge e g -> -nb_of_significant_degree g K - (VertexSet.union (interference_adj (fst_ext e) g) (interference_adj (snd_ext e) g)) - < K -> -conservative_coalescing_criteria e g K = true. - -Proof. -intros e g K H HHHH H0. -assert (In_graph (fst_ext e) g /\ In_graph (snd_ext e) g). -apply In_graph_edge_in_ext. assumption. destruct H1 as [HH HHH]. -unfold conservative_coalescing_criteria. -case_eq (is_precolored (fst_ext e) g && is_precolored (snd_ext e) g); intros. -destruct H. -case_eq (is_precolored (fst_ext e) g); intros. -elim H. apply (precolored_equiv _ _). split; assumption. -rewrite H2 in H1. inversion H1. -case_eq (is_precolored (snd_ext e) g); intros. -destruct H. apply (precolored_equiv _ _). split; assumption. -rewrite H2 in H1. destruct (is_precolored (fst_ext e)); auto. -destruct (le_lt_dec K (nb_of_significant_degree g K - (VertexSet.union (interference_adj (fst_ext e) g) - (interference_adj (snd_ext e) g)))). -intuition. -reflexivity. -Qed. - -Lemma conservative_coalescing_criteria_2 : forall e g palette, -conservative_coalescing_criteria e g palette = true -> -~VertexSet.In (fst_ext e) (precolored g) \/ -~VertexSet.In (snd_ext e) (precolored g). - -Proof. -intros. -unfold conservative_coalescing_criteria in H. -case_eq (is_precolored (fst_ext e) g && is_precolored (snd_ext e) g); intros. -rewrite H0 in H. -inversion H. -rewrite H0 in H. -case_eq (is_precolored (fst_ext e) g);intros. -right. -case_eq (is_precolored (snd_ext e) g); intros. -rewrite H1 in H0. rewrite H2 in H0. inversion H0. -intro H3. generalize (proj1 (precolored_equiv _ _) H3). intro. -rewrite H2 in H4. destruct H4. inversion H4. -left. -intro H4. generalize (proj1 (precolored_equiv _ _) H4). intro. -rewrite H1 in H2. destruct H2. inversion H2. -Qed. - -Lemma conservative_coalescing_criteria_3 : forall e g K, -conservative_coalescing_criteria e g K = true -> -nb_of_significant_degree g K - (VertexSet.union (interference_adj (fst_ext e) g) (interference_adj (snd_ext e) g)) - < K. - -Proof. -intros. -unfold conservative_coalescing_criteria in H. -case_eq (is_precolored (fst_ext e) g && is_precolored (snd_ext e) g); intros. -rewrite H0 in H. inversion H. -rewrite H0 in H. -destruct (le_lt_dec - K - (nb_of_significant_degree g K - (VertexSet.union (interference_adj (fst_ext e) g) - (interference_adj (snd_ext e) g)))). -inversion H. -assumption. -Qed. - -Definition is_none (o : option Edge.t) := -match o with -|None => true -|Some _ => false -end. - -(* Function picking an edge satisfying a function f in a set s*) -Definition get_element_such_f (f : Edge.t -> bool) s := -EdgeSet.fold (fun e o => if (is_none o) then if (f e) then Some e else None else o) s None. - -(* Specification of the get_element_such function *) -Lemma element_some : forall l (f : Edge.t -> bool) a, -fold_left - (fun (a : option Edge.t) (e : EdgeSet.elt) => - if is_none a then if f e then Some e else None else a) l (Some a) = Some a. - -Proof. -induction l; simpl; intros. -reflexivity. -apply IHl. -Qed. - -Lemma get_element_correct : forall f s (x : Edge.t), -get_element_such_f f s = Some x -> f x = true. - -Proof. -unfold get_element_such_f. intros f s x H. -unfold get_element_such_f in H. -rewrite EdgeSet.fold_1 in H. -induction (EdgeSet.elements s); simpl in H. -simpl in H. inversion H. -case_eq (f a); intro H0; rewrite H0 in H. -rewrite element_some in H. inversion H. subst. assumption. -apply IHl. assumption. -Qed. - -Lemma get_element_in : forall f s x, -get_element_such_f f s = Some x -> EdgeSet.In x s. - -Proof. -intros f s x H. -unfold get_element_such_f in H. -rewrite EdgeSet.fold_1 in H. -generalize (EdgeSet.elements_2);intro H0. -generalize (H0 s);clear H0;intro H0. -induction (EdgeSet.elements s);simpl in *. -inversion H. -case_eq (f a); intro H1; rewrite H1 in H. -rewrite element_some in H. inversion H. subst. -apply H0. intuition. -apply IHl; intuition. -Qed. - -Lemma compat_is_precolored : forall x y g, -Register.eq x y -> -is_precolored x g = is_precolored y g. - -Proof. -exact is_precolored_ext. -Qed. - -(* Definition of the any_coalescible_edge function which picks an affinity - edge satisfying the coalescing criterion for K in g in the set moves *) -Definition any_coalescible_edge moves g K := -match (get_element_such_f (fun e => conservative_coalescing_criteria e g K) moves) with -|None => None -|Some e => if (is_precolored (fst_ext e) g) then Some e else Some (snd_ext e, fst_ext e, get_weight e) -end. - -Lemma adj_of_significant_degree_compat_set : forall g s s' K, -VertexSet.Equal s s' -> -nb_of_significant_degree g K s = -nb_of_significant_degree g K s'. - -Proof. -intros g s s' K H. -unfold nb_of_significant_degree. -cut (eqlistA Register.eq (VertexSet.elements s) (VertexSet.elements s')). intro H0. -do 2 rewrite VertexSet.fold_1. -generalize H0. generalize (VertexSet.elements s'). -clear H0. generalize 0. -induction (VertexSet.elements s). simpl. -intros n l H0. -destruct l. -simpl. reflexivity. -inversion H0. -simpl. intro n. -destruct l0. simpl. intro H0. inversion H0. -intro H0. simpl. -inversion H0. subst. -case_eq (le_lt_dec K (interf_degree g a)); intros H1 _; -case_eq (le_lt_dec K (interf_degree g e)); intros H2 _. -apply IHl. assumption. -unfold interf_degree in H1. rewrite (compat_interference_adj _ _ _ H4) in H1. -apply False_ind. intuition. -unfold interf_degree in H2. apply False_ind. intuition. -unfold interf_degree in H1. rewrite (compat_interference_adj _ _ _ H4) in H1. -unfold interf_degree in H2. apply False_ind. intuition. -apply IHl. assumption. -apply SortA_equivlistA_eqlistA with (ltA := Register.lt). -apply Register.eq_refl. -apply Register.eq_sym. -apply Register.eq_trans. -apply Register.lt_trans. -apply Register.lt_not_eq. -apply OTFacts.lt_eq. -apply OTFacts.eq_lt. -apply VertexSet.elements_3. -apply VertexSet.elements_3. -apply equal_equivlist. assumption. -Qed. - -Lemma compat_criteria_aux : forall e e' g K, -eq e e' -> -In_graph_edge e g -> -conservative_coalescing_criteria e g K = true -> -conservative_coalescing_criteria e' g K = true. - -Proof. -intros e e' g K H HH H0. -apply conservative_coalescing_criteria_1. -generalize (conservative_coalescing_criteria_2 _ _ _ H0). intro H1. -destruct H1. -destruct (eq_charac _ _ H); change_rewrite; destruct H2. -left. rewrite <-H2. assumption. -right. rewrite <-H2. assumption. -destruct (eq_charac _ _ H); change_rewrite; destruct H2. -right. rewrite <-H3. assumption. -left. rewrite <-H3. assumption. - -rewrite <-H. assumption. - -generalize (conservative_coalescing_criteria_3 _ _ _ H0). intro H1. -destruct (eq_charac _ _ H); change_rewrite; destruct H2 as [H2 H3]; -rewrite <-(compat_interference_adj _ _ _ H2) in *; -rewrite <-(compat_interference_adj _ _ _ H3) in *. -assumption. -rewrite adj_of_significant_degree_compat_set with -(s':= VertexSet.union (interference_adj (fst_ext e) g) (interference_adj (snd_ext e) g)). -assumption. -apply union_sym. -Qed. - -Lemma compat_criteria : forall e e' g K, -eq e e' -> -In_graph_edge e g -> -conservative_coalescing_criteria e g K = -conservative_coalescing_criteria e' g K. - -Proof. -intros e e' g palette HH H. -case_eq (conservative_coalescing_criteria e g palette); intros. -symmetry. apply compat_criteria_aux with (e:=e). assumption. assumption. assumption. -case_eq (conservative_coalescing_criteria e' g palette); intros. -assert (conservative_coalescing_criteria e g palette = true). -apply compat_criteria_aux with (e:=e'). apply eq_sym. assumption. -rewrite <-HH. assumption. assumption. -rewrite H0 in H2. inversion H2. -reflexivity. -Qed. - -(* Specification of any_coalescible_edge *) -Lemma any_coalescible_edge_1 : forall e g K s, -(forall e', EdgeSet.In e' s -> In_graph_edge e' g) -> -any_coalescible_edge s g K = Some e -> -conservative_coalescing_criteria e g K = true /\ EdgeSet.In e s. - -Proof. -intros e g K s HH H. -unfold any_coalescible_edge in H. -case_eq (get_element_such_f (fun e : t => conservative_coalescing_criteria e g K) s). -intros t0 H0. -rewrite H0 in H. -destruct (is_precolored (fst_ext t0) g);inversion H;subst. -split. -apply (get_element_correct _ _ _ H0). -apply (get_element_in _ _ _ H0). -assert (eq t0 (snd_ext t0, fst_ext t0, get_weight t0)) by (Eq_comm_eq; apply Regs.eq_refl). -split. -rewrite (compat_criteria _ _ _ _ (eq_sym H1)). -apply (get_element_correct _ _ _ H0). -apply HH. rewrite <-H1. apply (get_element_in _ _ _ H0). -rewrite <-H1. apply (get_element_in _ _ _ H0). -intro H0. rewrite H0 in H. inversion H. -Qed. - -Lemma any_coalescible_edge_11 : forall e g palette s, -any_coalescible_edge s g palette = Some e -> -EdgeSet.In e s. - -Proof. -intros. -unfold any_coalescible_edge in H. -case_eq (get_element_such_f - (fun e : t => conservative_coalescing_criteria e g palette) s); intros. -rewrite H0 in H. -case_eq (is_precolored (fst_ext t0) g); intros. -rewrite H1 in H. -inversion H. subst. -apply (get_element_in _ _ _ H0). -rewrite H1 in H. -inversion H. subst. -rewrite edge_comm. change Regs.registers with Regs.t. rewrite <-(edge_eq t0). - apply (get_element_in _ _ _ H0). -rewrite H0 in H. -inversion H. -Qed. - -Lemma any_coalescible_edge_2 : forall e g palette s, -(forall e', EdgeSet.In e' s -> In_graph_edge e' g) -> -any_coalescible_edge s g palette = Some e -> -~VertexSet.In (snd_ext e) (precolored g). - -Proof. -intros e g palette s H H0. -intro H1. -generalize (any_coalescible_edge_1 _ _ _ _ H H0). intro HH. -unfold any_coalescible_edge in H0. -case_eq (get_element_such_f (fun e : t =>conservative_coalescing_criteria e g palette) s). -intros t0 H2. -rewrite H2 in H0. -case_eq (is_precolored (fst_ext t0) g);intro H3. -rewrite H3 in H0. -generalize (proj1 (precolored_equiv _ _ ) H1);clear H1;intro H1. -inversion H0;subst. destruct HH as [HH H5]. -generalize (conservative_coalescing_criteria_2 _ _ _ HH). intro H4. -destruct H4; elim H4; apply (proj2 (precolored_equiv _ _)). -split. assumption. apply (proj1 (In_graph_edge_in_ext _ _ (H e (get_element_in _ _ _ H2)))). -destruct H1. split; assumption. -rewrite H3 in H0. -inversion H0;subst. -clear H0. -change (snd_ext (snd_ext t0,fst_ext t0,get_weight t0)) with (fst_ext t0) in H1. -generalize (proj1 (precolored_equiv _ _) H1);clear H1;intro H1. -rewrite H3 in H1;inversion H1. inversion H0. -intro H2;rewrite H2 in H0. -inversion H0. -Qed. diff --git a/backend/Delete_Preference_Edges_Adjacency.v b/backend/Delete_Preference_Edges_Adjacency.v deleted file mode 100755 index 64ace15..0000000 --- a/backend/Delete_Preference_Edges_Adjacency.v +++ /dev/null @@ -1,60 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Merge_Degree. -Require Import Interference_adjacency. -Require Import Edges. -Require Import MyRegisters. - -Module Register := Regs. - -Import Edge RegFacts Props. - -(* The interference neighborhood of any vertex is left - unchanged when x is frozen *) -Lemma interf_adj_delete_preference : forall x r g H, -VertexSet.Equal (interference_adj x g) - (interference_adj x (delete_preference_edges r g H)). - -Proof. -split;intros. -rewrite in_interf. rewrite In_delete_preference_edges_edge. -rewrite in_interf in H0. split. assumption. -intro. destruct H1. -unfold aff_edge in H1. destruct H1. simpl in H1. congruence. - -rewrite in_interf in H0. rewrite In_delete_preference_edges_edge in H0. destruct H0. -rewrite in_interf. assumption. -Qed. - -(* The preference neighborhood of any vertex different from r - is obtained by removing r from its preference neighborhood in g *) -Lemma delete_preference_preference_adj : forall x r g H, -~Register.eq x r -> -VertexSet.Equal (preference_adj x (delete_preference_edges r g H)) - (VertexSet.remove r (preference_adj x g)). - -Proof. -intros. -split; intros. -apply VertexSet.remove_2. -intro. rewrite <-H2 in H1. -generalize (pref_adj_comm _ _ _ H1). intro. -rewrite in_pref in H3. destruct H3. -rewrite In_delete_preference_edges_edge in H3. -destruct H3. elim H4. split. -unfold aff_edge. exists x0. auto. -right. auto. -rewrite in_pref in H1. -destruct H1. rewrite In_delete_preference_edges_edge in H1. -destruct H1. rewrite in_pref. exists x0. assumption. - -(* <= *) -assert (~Register.eq r a). -intro. elim (VertexSet.remove_1 H2 H1). -generalize (VertexSet.remove_3 H1). clear H1. intro. -rewrite in_pref in H1. destruct H1. -rewrite in_pref. exists x0. -rewrite In_delete_preference_edges_edge. split. assumption. -intro. destruct H3. -destruct H4; change_rewrite. elim (H2 H4). elim H0; auto. -Qed. diff --git a/backend/Delete_Preference_Edges_Degree.v b/backend/Delete_Preference_Edges_Degree.v deleted file mode 100755 index 6b781f2..0000000 --- a/backend/Delete_Preference_Edges_Degree.v +++ /dev/null @@ -1,16 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Delete_Preference_Edges_Adjacency. -Require Import Edges. - -Import Edge Props RegFacts. - -(* The interference degree is left unchanged when r is frozen. Hence, - a vertex is of low-degree after freezing r iff it is before freezing r *) -Lemma delete_preference_edges_low : forall x r g K p, -has_low_degree g K x = has_low_degree (delete_preference_edges r g p) K x. - -Proof. -intros x r g K p. unfold has_low_degree, interf_degree. -rewrite <-(Equal_cardinal (interf_adj_delete_preference x r g p)). reflexivity. -Qed. diff --git a/backend/Delete_Preference_Edges_Move.v b/backend/Delete_Preference_Edges_Move.v deleted file mode 100755 index 66f9ab6..0000000 --- a/backend/Delete_Preference_Edges_Move.v +++ /dev/null @@ -1,138 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Merge_Move. -Require Import Edges. -Require Import MyRegisters. -Require Import Affinity_relation. -Require Import Interference_adjacency. -Require Import Delete_Preference_Edges_Adjacency. -Require Import Remove_Vertex_Move. - -Import Edge Props RegFacts. - -(* The frozen vertex is nonmove-related in the resulting graph *) -Lemma not_aff_related_delete_preference_edges : forall r g p, -move_related (delete_preference_edges r g p) r = false. - -Proof. -intros. apply move_related_false_charac2. -intros. -rewrite In_delete_preference_edges_edge in H0. destruct H0. -intro. elim H1. split; assumption. -Qed. - -(* A vertex which is move-related after a freeze is move-related before it *) -Lemma move_related_delete_move_related : forall x r g p, -move_related (delete_preference_edges r g p) x = true -> -move_related g x = true. - -Proof. -intros x r g p H. -generalize (move_related_charac _ _ H);intro H0. -destruct H0 as [y H0]. -destruct H0 as [H0 H1];destruct H1 as [H1 H2]. -rewrite In_delete_preference_edges_edge in H1. destruct H1. -apply (move_related_charac2 _ _ _ H0 H1 H2). -Qed. - -(* A vertex which is move-related in g and not in (delete_preference_edges r g H) - is a preference neighbor of r in g *) -Lemma delete_preference_edges_move_1 : forall x r g H, -~Register.eq r x -> -move_related g x = true -> -move_related (delete_preference_edges r g H) x = false -> -VertexSet.In x (preference_adj r g). - -Proof. -intros. generalize (not_move_related_empty_pref _ _ H2). intro. -rewrite delete_preference_preference_adj in H3. -destruct (In_dec r (preference_adj x g)). -apply pref_adj_comm. assumption. -assert (VertexSet.Equal (preference_adj x g) VertexSet.empty). -split; intros. -rewrite <-H3. apply VertexSet.remove_2. -intro. rewrite <-H5 in H4. elim n. auto. -assumption. -elim (VertexSet.empty_1 H4). -elim (move_related_not_empty_pref _ _ H1 H4). -auto. -Qed. - -(* A vertex which is move-related in g and nonmove-related in - (delete_preference_edges r g H) has a preference degree of 1 *) -Lemma delete_preference_edges_move_2 : forall x r g H, -~Register.eq r x -> -move_related g x = true -> -move_related (delete_preference_edges r g H) x = false -> -VertexSet.cardinal (preference_adj x g) = 1. - -Proof. -intros. generalize (not_move_related_empty_pref _ _ H2). intro. -rewrite delete_preference_preference_adj in H3. -destruct (In_dec r (preference_adj x g)). -cut (VertexSet.Equal (preference_adj x g) (VertexSet.singleton r)). intros. -rewrite H4. apply singleton_cardinal. -split; intros. -destruct (Register.eq_dec a r). -apply VertexSet.singleton_2. intuition. -assert (VertexSet.In a VertexSet.empty). -rewrite <-H3. apply VertexSet.remove_2; auto. -elim (VertexSet.empty_1 H5). -generalize (VertexSet.singleton_1 H4). intro. -rewrite <-H5. auto. -elim (move_related_not_empty_pref _ _ H1). -rewrite <-H3. - -split; intros. -rewrite Dec.F.remove_neq_iff. -assumption. -intro. rewrite <-H5 in H4. elim n. auto. -apply (VertexSet.remove_3 H4). -auto. -Qed. - -(* Reciprocally, a vertex which is move-related in g, - has a preference degree of 1 and is a preference neighbor of r - is nonmove-related in (delete_preference_edges r g H) *) -Lemma delete_preference_edges_move_inv : forall x r g H, -~Register.eq r x -> -move_related g x = true -> -VertexSet.In x (preference_adj r g) -> -VertexSet.cardinal (preference_adj x g) = 1 -> -move_related (delete_preference_edges r g H) x = false. - -Proof. -intros. -apply move_related_false_charac2. intros. -rewrite In_delete_preference_edges_edge in H5. -destruct H5. -intro. elim H6. -split. assumption. -cut (VertexSet.Equal (preference_adj x g) (VertexSet.singleton r)). intros. -destruct H7;[right|left]. -rewrite (edge_eq e) in H5. -assert (eq (x, snd_ext e, get_weight e) (fst_ext e, snd_ext e, get_weight e)). -Eq_eq. rewrite <-H9 in H5. -destruct H4. rewrite H4 in H5. -apply VertexSet.singleton_1. rewrite <-H8. -rewrite in_pref. exists x0. rewrite edge_comm. auto. -rewrite (edge_eq e) in H5. -assert (eq (fst_ext e, x, get_weight e) (fst_ext e, snd_ext e, get_weight e)). -Eq_eq. rewrite <-H9 in H5. -destruct H4. rewrite H4 in H5. -apply VertexSet.singleton_1. rewrite <-H8. -rewrite in_pref. exists x0. rewrite edge_comm. auto. -rewrite edge_comm. assumption. -apply cardinal_1_singleton. apply pref_adj_comm. assumption. assumption. -Qed. - -Lemma delete_preference_edges_move_false_false : forall x r g H, -move_related g x = false -> -move_related (delete_preference_edges r g H) x = false. - -Proof. -intros. -case_eq (move_related (delete_preference_edges r g H) x); intros. -rewrite (move_related_delete_move_related _ _ _ _ H1) in H0. inversion H0. -reflexivity. -Qed. diff --git a/backend/Edges.v b/backend/Edges.v deleted file mode 100755 index 1b57045..0000000 --- a/backend/Edges.v +++ /dev/null @@ -1,411 +0,0 @@ -Require Import FSets. -Require Import OrderedOption. -Require Import ZArith. -Require Import MyRegisters. - -(* Module of edges in a simple graph : there is never more - than one edge between two vertices *) - -Module Edge. - -Module Import Vertex := Regs. - -(* Definition of pair of vertices, to represent edges *) -Module VertexPair := PairOrderedType Vertex Vertex. - -(* N_as_OT is the OrderedType with t := N, - to describe weights of edges *) -Module OptionN_as_OT := OrderedOpt N_as_OT. - -(* An edge is finally a pair of endpoints and a weight *) -Module E := PairOrderedType VertexPair OptionN_as_OT. - -(* Useful modules imports *) -Module Import OTFacts := OrderedTypeFacts Vertex. -Module OTPairFacts := OrderedTypeFacts VertexPair. -Module OTEFacts := OrderedTypeFacts E. - -(* t is simply E.t *) -Definition t := E.t. - -(* accessors to the edges, their endpoints and their weight *) -Definition get_edge : t -> (Vertex.t*Vertex.t) := fun x => fst x. -Definition fst_ext : t -> Vertex.t := fun x => fst (get_edge x). -Definition snd_ext : t -> Vertex.t := fun x => snd (get_edge x). -Definition get_weight : t -> option N := fun x => snd x. - -(* get_edge e is only the extremities of e *) -Lemma get_edge_ext : forall e, -get_edge e = (fst_ext e,snd_ext e). - -Proof. -unfold fst_ext. unfold snd_ext. unfold get_edge. -simpl. intro e. destruct (fst e);auto. -Qed. - -(* An edge is the pair of vertices and a weight *) -Lemma edge_edge_weight : forall e, -e = (get_edge e, get_weight e). - -Proof. -unfold get_edge. unfold get_weight. -simpl. intro e. destruct e;auto. -Qed. - -(* Expansion of an edge *) -Lemma edge_eq : forall e, -e = (fst_ext e, snd_ext e, get_weight e). - -Proof. -intro e. unfold get_weight. -destruct e. unfold fst_ext. unfold snd_ext. unfold get_edge. simpl. -destruct p. auto. -Qed. - -(* Equality does not depend on the order of endpoints, e.g - (x,y,w) is equal to (y,x,w) so we need to define ordered edges *) -Definition ordered_edge e := -match (lt_dec (snd_ext e) (fst_ext e)) with -|left _ => (snd_ext e, fst_ext e, get_weight e) -|right _ => e -end. - -(* Commutativity of ordered_edge*) -Lemma ordered_edge_comm : forall x y o, -E.eq (ordered_edge (x,y,o)) (ordered_edge (y,x,o)). - -Proof. -unfold ordered_edge, snd_ext, fst_ext, get_edge. intros x y o. simpl. -destruct (lt_dec y x);destruct (lt_dec x y). -elim (lt_not_gt r r0). -apply E.eq_refl. -apply E.eq_refl. -destruct (Vertex.compare x y). -elim (n0 l). -unfold E.eq. simpl. repeat split; auto. -apply Vertex.eq_sym. auto. -apply OptionN_as_OT.eq_refl. -elim (n l). -Qed. - -(* Definition of equality as equality of s *) -Definition eq x y := E.eq (ordered_edge x) (ordered_edge y). - -(* The weak equality is the equality of their type (interference or preference) - and of their extremities, but not necessarily of their weight *) -Definition weak_eq x y := -(Vertex.eq (fst_ext x) (fst_ext y) /\ Vertex.eq (snd_ext x) (snd_ext y)) \/ -(Vertex.eq (fst_ext x) (snd_ext y) /\ Vertex.eq (snd_ext x) (fst_ext y)). - -(* Two edges having equal extremities (in the right order) and equal weights - are equal *) -Lemma eq_ordered_eq : forall x y, -E.eq x y -> eq x y. - -Proof. -unfold eq;unfold E.eq;intros x y H. -unfold ordered_edge;unfold fst_ext;unfold snd_ext;unfold get_edge;simpl. -destruct x as [x wx];destruct x as [x1 x2]; -destruct y as [y wy];destruct y as [y1 y2];simpl in *. -destruct (lt_dec x2 x1); destruct (lt_dec y2 y1);simpl in *. -intuition. -destruct H as [H HH];destruct H as [H H0]. -elim n. apply eq_lt with (y := x2). - apply Vertex.eq_sym;assumption. - apply lt_eq with (y := x1);assumption. -destruct H as [H HH];destruct H as [H H0]. -elim n. apply eq_lt with (y := y2). - assumption. - apply lt_eq with (y := y1); auto. apply Regs.eq_sym. auto. -assumption. -Qed. - -(* Definition of lt as the lexicographic order on endpoints - of the edges, after ordering *) -Definition lt x y := VertexPair.lt (get_edge (ordered_edge x)) (get_edge (ordered_edge y)) \/ - (VertexPair.eq (get_edge (ordered_edge x)) (get_edge (ordered_edge y)) /\ - OptionN_as_OT.lt (get_weight x) (get_weight y)). - -Lemma eq_refl : forall x, eq x x. - -Proof. -unfold eq;unfold E.eq. -repeat split;[apply Vertex.eq_refl|apply Vertex.eq_refl|apply OptionN_as_OT.eq_refl]. -Qed. - -Lemma eq_sym : forall x y, eq x y -> eq y x. - -Proof. -unfold eq;unfold E.eq;intros x y H. intuition. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -apply OptionN_as_OT.eq_sym;assumption. -Qed. - -Lemma eq_trans : forall x y z, eq x y -> eq y z -> eq x z. - -Proof. -unfold eq;intros x y z H H0. -apply (E.eq_trans _ _ _ H H0). -Qed. - -Lemma lt_trans : forall x y z, lt x y -> lt y z -> lt x z. - -Proof. -unfold lt;unfold get_edge;unfold ordered_edge;intros x y z H H0. -destruct (lt_dec (snd_ext x) (fst_ext x));simpl in *. -destruct (lt_dec (snd_ext y) (fst_ext y));simpl in *. -destruct (lt_dec (snd_ext z) (fst_ext z));simpl in *. -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (snd_ext y, fst_ext y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (snd_ext y, fst_ext y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (snd_ext y, fst_ext y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (snd_ext y, fst_ext y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -destruct (lt_dec (snd_ext y) (fst_ext y));simpl in *. -elim (n r0). -destruct (lt_dec (snd_ext z) (fst_ext z));simpl in *. -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (fst y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (fst y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (fst y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (fst y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -destruct (lt_dec (snd_ext y) (fst_ext y));simpl in *. -destruct (lt_dec (snd_ext z) (fst_ext z));simpl in *. -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (snd_ext y, fst_ext y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (snd_ext y,fst_ext y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (snd_ext y,fst_ext y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (snd_ext y, fst_ext y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -destruct (lt_dec (snd_ext y) (fst_ext y));simpl in *. -elim (n0 r). -destruct (lt_dec (snd_ext z) (fst_ext z));simpl in *. -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (fst y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (fst y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -destruct H;destruct H0. -left;apply (VertexPair.lt_trans _ _ _ H H0). -left;apply OTPairFacts.lt_eq with (y := (fst y));simpl. -unfold VertexPair.lt in H;simpl in H;assumption. -destruct H0 as [H0 HH0]. -unfold VertexPair.eq in H0;assumption. -left;apply OTPairFacts.eq_lt with (y := (fst y));simpl. -destruct H as [H HH]. -unfold VertexPair.eq in H;unfold VertexPair.lt in H0; -simpl in H0;assumption. -unfold VertexPair.lt in H0;simpl in H0;assumption. -right;destruct H as [H HH];destruct H0 as [H0 HH0];split. -apply (VertexPair.eq_trans _ _ _ H H0). -apply (OptionN_as_OT.lt_trans _ _ _ HH HH0). -Qed. - -Lemma weight_ordered_weight : forall x, -get_weight x = get_weight (ordered_edge x). - -Proof. -intro x;unfold ordered_edge;unfold get_weight;simpl. -destruct (lt_dec (snd_ext x) (fst_ext x));simpl;reflexivity. -Qed. - -Lemma lt_not_eq : forall x y, lt x y -> ~eq x y. - -Proof. -unfold lt;unfold eq;unfold E.eq;intros x y H. -destruct H. -generalize (VertexPair.lt_not_eq _ _ H);intro H0. -unfold VertexPair.eq in H0. -intro H1; elim H0. -destruct H1 as [H1 H2]. -unfold get_edge;assumption. -destruct H as [H H0]. -unfold VertexPair.eq in H. -unfold get_edge in H. -intro H1. -destruct H1 as [H1 HH1]. -elim (OptionN_as_OT.lt_not_eq _ _ H0). -rewrite (weight_ordered_weight x). -rewrite (weight_ordered_weight y). -unfold get_weight;assumption. -Qed. - -Lemma compare : forall x y, Compare lt eq x y. - -Proof. -intros x y. -destruct (OTEFacts.lt_dec (ordered_edge x) (ordered_edge y)). -apply LT. -unfold lt;unfold get_edge;unfold VertexPair.lt; -unfold VertexPair.eq. -rewrite (weight_ordered_weight x). -rewrite (weight_ordered_weight y). -unfold get_weight. -destruct (ordered_edge x) as [ex wx];destruct ex as [ex1 ex2]; -destruct (ordered_edge y) as [ey wy];destruct ey as [ey1 ey2];simpl in *. -assumption. -destruct (OTEFacts.eq_dec (ordered_edge x) (ordered_edge y)). -apply EQ. -unfold eq. -destruct (ordered_edge x) as [ex wx];destruct ex as [ex1 ex2]; -destruct (ordered_edge y) as [ey wy];destruct ey as [ey1 ey2];simpl in *. -destruct a as [H H0];destruct H as [H H1]. -unfold E.eq;auto. -apply GT. -generalize (OTEFacts.le_neq n n0);intro H. -unfold lt;unfold get_edge;unfold VertexPair.lt; -unfold VertexPair.eq. -rewrite (weight_ordered_weight x). -rewrite (weight_ordered_weight y). -unfold get_weight. -destruct (ordered_edge x) as [ex wx];destruct ex as [ex1 ex2]; -destruct (ordered_edge y) as [ey wy];destruct ey as [ey1 ey2];simpl in *. -assumption. -Qed. - -(* Edges commutativity *) -Lemma edge_comm : forall x y o, eq (x,y,o) (y,x,o). - -Proof. -unfold eq;apply ordered_edge_comm. -Qed. - -(* Definition of affinity edges *) -Definition aff_edge x := exists w, get_weight x = Some w. - -(* Definition of interference edges *) -Definition interf_edge x := get_weight x = None. - -(* An edge is incident to a vertex iff this vertex is an endpoint of the edge *) -Definition incident e x := Vertex.eq x (fst_ext e) \/ Vertex.eq x (snd_ext e). - -(* An edge is incident to a vertex, or is not *) -Lemma incident_dec : forall e x, -incident e x \/ ~incident e x. - -Proof. -intros e x. -destruct (eq_dec x (fst_ext e)). -left;unfold incident;left;assumption. -destruct (eq_dec x (snd_ext e)). -left;unfold incident;right;assumption. -right;unfold incident;intro Heq. -destruct Heq;[elim (n H)|elim (n0 H)]. -Qed. - -(* Definition redirect : redirect x y e replaces the extremity x of the edge e - with y, if e is incident to x *) -Definition redirect x y e := -match eq_dec (fst_ext e) x with -|left _ => (y, snd_ext e, get_weight e) -|right _ => match eq_dec (snd_ext e) x with - | left _ => (fst_ext e, y, get_weight e) - | right _ => e - end -end. - -(* Decidable equality over edges *) -Lemma eq_dec : forall x y, {eq x y}+{~eq x y}. - -Proof. -intros x y. -destruct (compare x y). -right. apply lt_not_eq. assumption. -left. assumption. -right. intro H. generalize (eq_sym _ _ H). intro H0. elim (lt_not_eq _ _ l H0). -Qed. - -(* Equality of edges implies constraints of their endpoints *) -Lemma eq_charac : forall x y, -eq x y -> (Vertex.eq (fst_ext x) (fst_ext y) /\ Vertex.eq (snd_ext x) (snd_ext y)) \/ - (Vertex.eq (fst_ext x) (snd_ext y) /\ Vertex.eq (snd_ext x) (fst_ext y)). - -Proof. -intros x y H;unfold eq in H;unfold ordered_edge in H; -unfold get_edge in H. -destruct (lt_dec (snd_ext x) (fst_ext x)); -destruct (lt_dec (snd_ext y) (fst_ext y)); -unfold E.eq in H;simpl in H;intuition. -Qed. - -Definition same_type e e' := (aff_edge e /\ aff_edge e') \/ - (interf_edge e /\ interf_edge e'). - -End Edge.
\ No newline at end of file diff --git a/backend/EqualSetMap.v b/backend/EqualSetMap.v deleted file mode 100755 index 292b1c0..0000000 --- a/backend/EqualSetMap.v +++ /dev/null @@ -1,82 +0,0 @@ -Require Import List. -Require Import InterfGraphMapImp. - -Section fold_assoc_interf_map. - -Variable A : Type. - -Inductive eq_set_option : option VertexSet.t -> option VertexSet.t -> Prop := -|None_eq : eq_set_option None None -|Some_eq : forall s s', VertexSet.Equal s s' -> eq_set_option (Some s) (Some s'). - -Definition EqualSetMap m1 m2 := forall x, eq_set_option (VertexMap.find x m1) (VertexMap.find x m2). - -Lemma EqualSetMap_refl : forall m, EqualSetMap m m. - -Proof. -unfold EqualSetMap. intro m. intro x. -destruct (VertexMap.find x m). -constructor. intuition. -constructor. -Qed. - -Lemma EqualSetMap_trans : forall m1 m2 m3, -EqualSetMap m1 m2 -> -EqualSetMap m2 m3 -> -EqualSetMap m1 m3. - -Proof. -intros m1 m2 m3 H H0. -unfold EqualSetMap in *. -intro x. -generalize (H x). clear H. intro H. -generalize (H0 x). clear H0. intro H0. -destruct (VertexMap.find x m1). -inversion H. subst. -rewrite <-H2 in H0. -destruct (VertexMap.find x m3). -constructor. inversion H0. subst. -rewrite H3. assumption. -inversion H0. -destruct (VertexMap.find x m3). -inversion H0. inversion H. subst. rewrite <-H4 in H1. inversion H1. -constructor. -Qed. - -Lemma fold_left_compat_map : forall (f : VertexMap.t VertexSet.t -> A -> VertexMap.t VertexSet.t) l e e', -EqualSetMap e e' -> -(forall e1 e2 a, EqualSetMap e1 e2 -> EqualSetMap (f e1 a) (f e2 a)) -> -EqualSetMap (fold_left f l e) (fold_left f l e'). - -Proof. -intro f;induction l;simpl. -auto. -intros e e' H H0 H1. -apply (IHl (f e a) (f e' a)). -apply H0;assumption. -assumption. -Qed. - -Lemma fold_left_assoc_map : forall l (f : VertexMap.t VertexSet.t -> A -> VertexMap.t VertexSet.t) x h, -(forall (y z : A) s, EqualSetMap (f (f s y) z) (f (f s z) y)) -> -(forall e1 e2 a, EqualSetMap e1 e2 -> EqualSetMap (f e1 a) (f e2 a)) -> -EqualSetMap (fold_left f (h :: l) x) (f (fold_left f l x) h). - -Proof. -induction l;simpl;intros f x h H H0. -apply EqualSetMap_refl. -apply EqualSetMap_trans with (m2 := fold_left f (h :: l) (f x a)). -simpl. apply fold_left_compat_map. apply H. -assumption. -apply IHl. assumption. assumption. -Qed. -(* -Add Morphism EqualSetMap : equalsetmap_m. - -Proof. -unfold EqualSetMap, VertexMap.Equal. split; intros. -rewrite <-(H0 x1). rewrite <-(H x1). apply H1. -rewrite (H0 x1). rewrite (H x1). apply H1. -Qed. -*) -End fold_assoc_interf_map.
\ No newline at end of file diff --git a/backend/Freeze_WL.v b/backend/Freeze_WL.v deleted file mode 100755 index a235374..0000000 --- a/backend/Freeze_WL.v +++ /dev/null @@ -1,210 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Merge_WL. -Require Import Edges. -Require Import Remove_Vertex_WL. -Require Import IRC_graph. -Require Import Delete_Preference_Edges_Degree. -Require Import WS. -Require Import Delete_Preference_Edges_Move. -Require Import Graph_Facts. -Require Import Interference_adjacency. -Require Import Affinity_relation. - -Import RegFacts Props. - -Definition delete_preferences_wl2 v ircg k := -let wl := irc_wl ircg in -let g := irc_g ircg in -let simplify := get_simplifyWL wl in -let freeze := get_freezeWL wl in -let spill := get_spillWL wl in -let moves := get_movesWL wl in -let adj_fst := VertexSet.diff (preference_adj v g) (precolored g) in -let new_simp := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g k x) adj_fst in -let simplifyWL' := VertexSet.add v (VertexSet.union simplify new_simp) in -let freezeWL' := VertexSet.remove v (VertexSet.diff freeze new_simp) in -let movesWL' := not_incident_edges v moves g in -(spill, freezeWL', simplifyWL', movesWL'). - -Lemma WS_delete_preferences_wl2 : forall r ircg (Hin : In_graph r (irc_g ircg)), -VertexSet.In r (get_freezeWL (irc_wl ircg)) -> -WS_properties (delete_preference_edges r (irc_g ircg) Hin) - (VertexSet.cardinal (pal ircg)) - (delete_preferences_wl2 r ircg (VertexSet.cardinal (pal ircg))). - - -Proof. -intros r ircg Hin HH. -unfold WS_properties. unfold delete_preferences_wl2. -generalize (HWS_irc ircg). intro HWS. -set (wl := irc_wl ircg) in *. -set (g := irc_g ircg) in *. -set (simplify := get_simplifyWL wl) in *. -set (freeze := get_freezeWL wl) in *. -set (spill := get_spillWL wl) in *. -set (adj_fst := VertexSet.diff (preference_adj r g) (precolored g)) in *. -set (new_simp := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g))) adj_fst) in *. -set (simplifyWL_ := VertexSet.union simplify new_simp) in *. -set (simplifyWL' := VertexSet.add r simplifyWL_) in *. -set (freezeWL' := VertexSet.diff freeze new_simp) in *. -set (movesWL' := not_incident_edges r (get_movesWL wl) g) in *. -rewrite <-(Hk ircg) in HWS. set (K := VertexSet.cardinal (pal ircg)) in *. - -unfold get_spillWL, get_simplifyWL, get_freezeWL, get_movesWL. simpl. - -split. intro x. -rewrite <-delete_preference_edges_low. -rewrite precolored_delete_preference_edges. -rewrite In_delete_preference_edges_vertex. -split; intros. -apply (In_spill_props _ _ _ _ _ _ _ _ H (refl_equal _) HWS). -WS_apply HWS. assumption. - -(* freeze *) -split. intro x. -rewrite precolored_delete_preference_edges. -rewrite <-delete_preference_edges_low. - -split; intros. -unfold freezeWL' in H. -assert (~Register.eq r x) as Hneq. -intro. elim (VertexSet.remove_1 H0 H). -generalize (VertexSet.remove_3 H). clear H. intro H. -generalize (VertexSet.diff_1 H). generalize (VertexSet.diff_2 H). clear H. intros. -generalize (In_freeze_props _ _ _ _ _ _ _ _ H0 (refl_equal _) HWS). intro. -destruct H1. destruct H2. destruct H3. -split. -assumption. -split. - -destruct (Register.eq_dec r x). elim (Hneq e). -case_eq (move_related (delete_preference_edges r g Hin) x); intros. -reflexivity. -elim H. unfold new_simp. -apply VertexSet.filter_3. -apply compat_move_up. -apply VertexSet.diff_3. -apply delete_preference_edges_move_1 with (H := Hin); auto. -assumption. -rewrite andb_true_iff. split. -apply eq_K_1. -apply delete_preference_edges_move_2 with (H := Hin); auto. -assumption. -assumption. - -destruct H. destruct H0. -apply VertexSet.remove_2. intro. -rewrite (compat_bool_move _ _ _ (Register.eq_sym _ _ H2)) in H0. -rewrite (not_aff_related_delete_preference_edges r g Hin) in H0. inversion H0. -unfold freezeWL'. apply VertexSet.diff_3. -WS_apply HWS. split. -assumption. -split. -apply (move_related_delete_move_related _ _ _ _ H0). -assumption. -intro. assert (move_related (delete_preference_edges r g Hin) x = false). -apply delete_preference_edges_move_inv. -intro. -rewrite (compat_bool_move _ _ _ (Register.eq_sym _ _ H3)) in H0. -rewrite (not_aff_related_delete_preference_edges r g Hin) in H0. inversion H0. -apply (move_related_delete_move_related _ _ _ _ H0). -unfold new_simp in H2. -apply (VertexSet.diff_1 (VertexSet.filter_1 (compat_move_up _ _) H2)). -symmetry. apply eq_K_2. generalize (VertexSet.filter_2 (compat_move_up _ _) H2). intro. -rewrite andb_true_iff in H3. destruct H3. assumption. -rewrite H0 in H3. inversion H3. - -(* simplify *) -split. intro x. -rewrite <-delete_preference_edges_low. -rewrite precolored_delete_preference_edges. -rewrite In_delete_preference_edges_vertex. -split; intros. -unfold simplifyWL' in H. -destruct (proj1 (Dec.F.add_iff _ _ _) H). -generalize (In_freeze_props _ _ _ _ _ _ _ _ HH (refl_equal _) HWS). intro. -destruct H1. destruct H2. destruct H3. -split. rewrite <-(compat_bool_low _ _ _ _ H0). assumption. -split. rewrite <-(compat_bool_move _ _ _ H0). apply not_aff_related_delete_preference_edges. -split. rewrite <-H0. assumption. -rewrite <-H0. assumption. - -unfold simplifyWL_ in H0. -destruct (VertexSet.union_1 H0). -generalize (In_simplify_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS). intro. -destruct H2. destruct H3. destruct H4. -split. -assumption. -split. -apply delete_preference_edges_move_false_false. assumption. -split; assumption. -unfold new_simp in H1. -generalize (VertexSet.filter_1 (compat_move_up _ _) H1). intro. -generalize (VertexSet.filter_2 (compat_move_up _ _) H1). clear H1. intro. -rewrite andb_true_iff in H1. destruct H1. -split. assumption. -split. apply delete_preference_edges_move_inv. -intro. elim (not_in_pref_self r g). -rewrite <-H4 in H2. apply (VertexSet.diff_1 H2). -apply move_related_card. -unfold pref_degree. rewrite <-(eq_K_2 _ _ H1). auto. -apply (VertexSet.diff_1 H2). -symmetry. apply (eq_K_2 _ _ H1). -split. -apply (in_pref_in _ _ _ (VertexSet.diff_1 H2)). -apply (VertexSet.diff_2 H2). - -destruct H. destruct H0. destruct H1. -destruct (Register.eq_dec r x). -apply VertexSet.add_1. auto. -apply VertexSet.add_2. -case_eq (In_dec x adj_fst); intros. -case_eq (eq_K 1 (VertexSet.cardinal (preference_adj x g))); intros. -apply VertexSet.union_3. apply VertexSet.filter_3. -apply compat_move_up. -assumption. -rewrite H4. rewrite H. auto. -apply VertexSet.union_2. WS_apply HWS. -split. assumption. -split. -case_eq (move_related g x); intros. -generalize (delete_preference_edges_move_2 _ _ _ Hin n H5 H0). intro. -rewrite (eq_K_1 _ _ H6) in H4. inversion H4. -reflexivity. -split; assumption. -apply VertexSet.union_2. -WS_apply HWS. -split. assumption. -split. case_eq (move_related g x); intros. -generalize (delete_preference_edges_move_1 _ _ _ Hin n H4 H0). intro. -elim n0. apply VertexSet.diff_3; assumption. -reflexivity. -split; assumption. - -unfold movesWL'. intro. rewrite not_incident_edges_1. -rewrite In_delete_preference_edges_edge. -split; intros. -destruct H. -generalize (In_move_props _ _ _ _ _ _ _ _ H (refl_equal _) HWS). intro. -destruct H1. -split. assumption. -split. assumption. -intro. elim H0. destruct H3. assumption. -destruct H. destruct H0. -split. WS_apply HWS. split; assumption. -intro. elim H1. split; assumption. - -intros. apply (In_move_props _ _ _ _ _ _ _ _ H (refl_equal _) HWS). -Qed. - -Lemma WS_freeze : forall r ircg (Hin : In_graph r (irc_g ircg)), -VertexSet.In r (get_freezeWL (irc_wl ircg)) -> -WS_properties (delete_preference_edges r (irc_g ircg) Hin) - (irc_k ircg) - (delete_preferences_wl2 r ircg (irc_k ircg)). - - -Proof. -intros. rewrite <-(Hk ircg). apply WS_delete_preferences_wl2. auto. -Qed. diff --git a/backend/Graph_Facts.v b/backend/Graph_Facts.v deleted file mode 100755 index 3919b1f..0000000 --- a/backend/Graph_Facts.v +++ /dev/null @@ -1,191 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import ZArith. -Require Import Edges. -Require Import MyRegisters. - -Import Edge OTFacts. -Module Import RegFacts := Facts VertexSet. -Module Import RegRegFacts := Facts EdgeSet. - -(* Definition of useful morphisms *) - -Add Morphism In_graph : In_graph_m. - -Proof. -unfold In_graph;intros x y H g. -split;intro H0;[rewrite <-H|rewrite H];assumption. -Qed. - -Add Morphism In_graph_edge : In_graph_edge_m. - -Proof. -unfold In_graph_edge; intros x y H g. fold (eq x y) in H. -split;intro H0;[rewrite <-H|rewrite H];assumption. -Qed. - -Add Morphism move_related : move_related_m. - -Proof. -unfold move_related. intros. rewrite (compat_preference_adj _ _ _ H). reflexivity. -Qed. - -Add Morphism get_weight : get_weight_m. - -Proof. -intros x y H. -rewrite (weight_ordered_weight x);rewrite (weight_ordered_weight y). -unfold get_weight;unfold E.eq in H. -destruct H as [_ H];inversion H;[|rewrite H2];reflexivity. -Qed. - -(* Useful rewriting lemmas *) - -Lemma rewrite_fst : forall x y z, -fst_ext (x,y,z) = x. - -Proof. -auto. -Qed. - -Lemma rewrite_snd : forall x y z, -snd_ext (x,y,z) = y. - -Proof. -auto. -Qed. - -Lemma rewrite_weight : forall x y z, -get_weight (x,y,z) = z. - -Proof. -auto. -Qed. - -(* A rewriting tactic *) - -Ltac change_rewrite := -repeat (try rewrite rewrite_fst in *; - try rewrite rewrite_snd in *; - try rewrite rewrite_weight in *). - -(* Two tactics for proving equality of edges *) -Ltac Eq_eq := -apply eq_ordered_eq;unfold E.eq; -split;[simpl;split;auto; intuition |try apply OptionN_as_OT.eq_refl; intuition]. - -Ltac Eq_comm_eq := rewrite edge_comm; Eq_eq. - -(* Extensionnality of redirect *) -Lemma redirect_ext : forall e e' x y, -eq e e' -> -eq (redirect x y e) (redirect x y e'). - -Proof. -intros e e' x y H. -destruct e as [xe we];destruct xe as [ex1 ex2]; -destruct e' as [xe' we'];destruct xe' as [e'x1 e'x2];simpl in *. -generalize (get_weight_m _ _ H);simpl;intro Heq;subst. -destruct (eq_charac _ _ H);destruct H0 as [H0 H1];unfold redirect;change_rewrite. -destruct (OTFacts.eq_dec ex1 x). -destruct (OTFacts.eq_dec e'x1 x). -Eq_eq. intuition. -elim (n (Regs.eq_trans (Regs.eq_sym H0) r)). -destruct (OTFacts.eq_dec e'x1 x). -elim (n (Regs.eq_trans H0 r)). -destruct (OTFacts.eq_dec ex2 x). -destruct (OTFacts.eq_dec e'x2 x). -Eq_eq. intuition. -elim (n1 (Regs.eq_trans (Regs.eq_sym H1) r)). -destruct (OTFacts.eq_dec e'x2 x). -elim (n1 (Regs.eq_trans H1 r)). -Eq_eq. -destruct (OTFacts.eq_dec ex1 x). -destruct (OTFacts.eq_dec e'x1 x). -Eq_eq. intuition. -apply (Regs.eq_trans H1 (Regs.eq_trans r0 (Regs.eq_trans (Regs.eq_sym r) H0))). -destruct (OTFacts.eq_dec e'x2 x). -Eq_comm_eq. intuition. -elim (n0 (Regs.eq_trans (Regs.eq_sym H0) r)). -destruct (OTFacts.eq_dec e'x2 x). -elim (n (Regs.eq_trans H0 r)). -destruct (OTFacts.eq_dec ex2 x). -destruct (OTFacts.eq_dec e'x1 x). -Eq_comm_eq. -elim (n1 (Regs.eq_trans (Regs.eq_sym H1) r)). -destruct (OTFacts.eq_dec e'x1 x). -elim (n1 (Regs.eq_trans H1 r)). -Eq_comm_eq. -Qed. - -(* The weight is left unchanged while applying redirect *) -Lemma redirect_weight_eq : forall e x y, -Edge.get_weight (redirect x y e) = Edge.get_weight e. - -Proof. -unfold redirect;intros e x y;destruct e as [e w];destruct e as [ex1 ex2];change_rewrite. -destruct (OTFacts.eq_dec ex1 x);destruct (OTFacts.eq_dec ex2 x);reflexivity. -Qed. - -(* Specification of redirect *) -Lemma redirect_charac : forall e x y, -(eq e (redirect x y e) /\ (~Regs.eq x (fst_ext e) /\ ~Regs.eq x (snd_ext e))) \/ -(eq (y, snd_ext e, get_weight e) (redirect x y e) /\ Regs.eq x (fst_ext e)) \/ -(eq (fst_ext e, y, get_weight e) (redirect x y e) /\ Regs.eq x (snd_ext e)). - -Proof. -intros e x y. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e) x). -right. left. split. apply eq_refl. auto. -destruct (OTFacts.eq_dec (snd_ext e) x). -right. right. split. apply eq_refl. auto. -left. split. apply eq_refl. split; intuition. -Qed. - -(* Weak equality implies equality for interference edges *) -Lemma weak_eq_interf_eq : forall x y, -weak_eq x y -> -interf_edge x -> -interf_edge y -> -eq x y. - -Proof. -unfold weak_eq, interf_edge, get_weight. intros. destruct H; destruct H. -Eq_eq. rewrite H0. rewrite H1. apply OptionN_as_OT.eq_refl. -rewrite (edge_eq x). rewrite (edge_eq y). -Eq_comm_eq. simpl. unfold get_weight. -rewrite H0. rewrite H1. apply OptionN_as_OT.eq_refl. -Qed. - -(* The second endpoint of e does not belongs to (merge e g H H0) *) -Lemma not_in_merge_snd : forall e g H H0, -~In_graph (snd_ext e) (merge e g H H0). - -Proof. -intros. intro. -rewrite In_merge_vertex in H1. destruct H1. -elim (H2 (Regs.eq_refl _)). -Qed. - -(* Extensionnality of the low-degree function *) -Lemma compat_bool_low : forall g K, -compat_bool Regs.eq (has_low_degree g K). - -Proof. -unfold compat_bool, has_low_degree, interf_degree. intros. -rewrite (compat_interference_adj _ _ _ H). reflexivity. -Qed. - -(* There cannot exist both an interference and - a preference between two vertices *) -Lemma interf_pref_conflict : forall x y g, -Prefere x y g /\ Interfere x y g -> False. - -Proof. -unfold Prefere, Interfere. intros. destruct H. destruct H. -assert (eq (x,y,Some x0) (x,y,None)). -apply is_simple_graph with (g:=g); auto. -unfold weak_eq. left. change_rewrite. split; auto. -generalize (get_weight_m _ _ H1). simpl. congruence. -Qed. diff --git a/backend/Graph_translation.v b/backend/Graph_translation.v deleted file mode 100644 index fb8f6c2..0000000 --- a/backend/Graph_translation.v +++ /dev/null @@ -1,3552 +0,0 @@ -Require Import InterfGraph. -Require Import InterfGraphMapImp. -Require Import MyRegisters. -Require Import RTLtyping. -Require Import NArith. -Require Import List. -Require Import InterfGraph_Construction. -Require Import Edges. -Require Import EqualSetMap. - -Import Edge. - -Definition new_graph := tt. - -Section Translation. - -Variable g : graph. -Variable interfgraph_vertices : VertexSet.t. - -Definition IE_reg_reg := -SetRegReg.fold (fun e s => EdgeSet.add (Regs.reg_to_Reg (fst e), Regs.reg_to_Reg (snd e), None) s) - (interf_reg_reg g) EdgeSet.empty. - -Definition IE_reg_mreg := -SetRegMreg.fold (fun e s => EdgeSet.add (Regs.reg_to_Reg (fst e), Regs.mreg_to_Reg (snd e), None) s) - (interf_reg_mreg g) EdgeSet.empty. - -Definition AE_reg_reg := -SetRegReg.fold (fun e s => EdgeSet.add (Regs.reg_to_Reg (fst e), Regs.reg_to_Reg (snd e), Some N0) s) - (pref_reg_reg g) EdgeSet.empty. - -Definition AE_reg_mreg := -SetRegMreg.fold (fun e s => EdgeSet.add (Regs.reg_to_Reg (fst e), Regs.mreg_to_Reg (snd e), Some N0) s) - (pref_reg_mreg g) EdgeSet.empty. - -Definition interfgraph_interference_edges := EdgeSet.union IE_reg_reg IE_reg_mreg. - -Definition resolve_conflicts s1 s2 := -EdgeSet.fold (fun e s => if EdgeSet.mem (fst_ext e, snd_ext e,None) s2 - then s else EdgeSet.add e s) s1 EdgeSet.empty. - -Definition interfgraph_affinity_edges := -resolve_conflicts -(EdgeSet.union AE_reg_reg AE_reg_mreg) -interfgraph_interference_edges. - -Hypothesis extremities_interf_graph : -forall e, EdgeSet.In e interfgraph_affinity_edges \/ - EdgeSet.In e interfgraph_interference_edges -> - VertexSet.In (fst_ext e) interfgraph_vertices /\ - VertexSet.In (snd_ext e) interfgraph_vertices. - -Definition new_adj_set x y m := -match (VertexMap.find x m) with - | None => VertexSet.singleton y - | Some s => VertexSet.add y s -end. - -Definition empty_map := -VertexSet.fold - (fun x m => VertexMap.add x VertexSet.empty m) - interfgraph_vertices - (VertexMap.empty VertexSet.t). - -Definition set2map s map := -EdgeSet.fold (fun e m => VertexMap.add (fst_ext e) (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) (new_adj_set (snd_ext e) (fst_ext e) m) m)) - s map. - -Definition interf_map := set2map interfgraph_interference_edges empty_map. -Definition pref_map := set2map interfgraph_affinity_edges empty_map. - -Lemma add_simpl : forall l a s s2, -EdgeSet.Equal - (fold_left - (fun (a0 : EdgeSet.t) (e0 : EdgeSet.elt) => - if EdgeSet.mem (fst_ext e0, snd_ext e0, None) s2 - then a0 - else EdgeSet.add e0 a0) l (EdgeSet.add a s)) -(EdgeSet.add a -(fold_left - (fun (a0 : EdgeSet.t) (e0 : EdgeSet.elt) => - if EdgeSet.mem (fst_ext e0, snd_ext e0, None) s2 - then a0 - else EdgeSet.add e0 a0) l s)). - -Proof. -induction l;simpl;intros. -intuition. -case_eq (EdgeSet.mem (fst_ext a,snd_ext a,None) s2);intro H. -apply IHl. -do 3 rewrite IHl. -apply RegRegProps.add_add. -Qed. - - -Lemma resolve_conflicts_2 : forall e s1 s2, -EdgeSet.In e (resolve_conflicts s1 s2) -> -EdgeSet.In e s1 /\ ~EdgeSet.In (fst_ext e,snd_ext e, None) s2. - -Proof. -intros e s1 s2 H. -unfold resolve_conflicts in H. -rewrite EdgeSet.fold_1 in H. -split. -generalize (EdgeSet.elements_2);intro H0;generalize (H0 s1);clear H0;intro H0. -induction (EdgeSet.elements s1);simpl in *. -elim (EdgeSet.empty_1 H). -destruct (eq_dec a e). -apply H0. -fold (eq a e) in e0. -generalize (Edge.eq_sym e0);clear e0;intro e0. -intuition. -case_eq (EdgeSet.mem (fst_ext a, snd_ext a,None) s2);intro H1;rewrite H1 in H. -apply IHl. -apply H. -intuition. -apply IHl. - -rewrite add_simpl in H. -apply (EdgeSet.add_3 n H). -intuition. -induction (EdgeSet.elements s1);simpl in *. -elim (EdgeSet.empty_1 H). -case_eq (EdgeSet.mem (fst_ext a,snd_ext a,None) s2);intro H0;rewrite H0 in H. -apply IHl;assumption. -destruct (eq_dec a e). -intro H1. -destruct (eq_charac _ _ e0). -destruct H2 as [H2 HH2]. -assert (eq (fst_ext a,snd_ext a,None) (fst_ext e,snd_ext e,None)) by Eq_eq. -rewrite (RegRegProps.Dec.F.mem_b s2 H3) in H0. -generalize (EdgeSet.mem_1 H1);clear H1;intro H1. -rewrite H0 in H1;inversion H1. -destruct H2 as [H2 HH2]. -assert (eq (fst_ext a,snd_ext a,None) (fst_ext e,snd_ext e,None)) by Eq_comm_eq. -rewrite (RegRegProps.Dec.F.mem_b s2 H3) in H0. -generalize (EdgeSet.mem_1 H1);clear H1;intro H1. -rewrite H0 in H1;inversion H1. -apply IHl. -rewrite add_simpl in H. -apply (EdgeSet.add_3 n H). -Qed. - -(* -intros g e H. -unfold interfgraph_vertices. -destruct H. -unfold interfgraph_affinity_edges in H. -generalize (proj1 (resolve_conflicts_2 _ _ _ H)). clear H. intro H. -destruct (EdgeSet.union_1 H). -cut (VertexSet.In (fst_ext e) (V_pref_reg_reg g) /\ - VertexSet.In (snd_ext e) (V_pref_reg_reg g)). -intro H1. destruct H1 as [H1 H2]. split; -(do 2 apply VertexSet.union_3; -apply VertexSet.union_2; assumption). -unfold V_pref_reg_reg. -unfold AE_reg_reg in H0. -rewrite SetRegReg.fold_1 in *. -induction (SetRegReg.elements (pref_reg_reg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.reg_to_Reg (snd a), Some N0) e) in H1. -rewrite RegFacts.fold_left_assoc. -destruct (eq_charac _ _ H1); destruct H2 as [H2 H3]. -change_rewrite. split. -apply VertexSet.add_1. assumption. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -change_rewrite. split. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -apply VertexSet.add_1. assumption. -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.reg_to_Reg (snd y)) s) - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.reg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.reg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -rewrite RegFacts.fold_left_assoc. -split; do 2 apply VertexSet.add_2. -apply (proj1 (IHl H1)). -apply (proj2 (IHl H1)). -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.reg_to_Reg (snd y)) s) - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.reg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.reg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. - -cut (VertexSet.In (fst_ext e) (V_pref_reg_mreg g) /\ - VertexSet.In (snd_ext e) (V_pref_reg_mreg g)). -intro H1. destruct H1 as [H1 H2]. split; -(do 3 apply VertexSet.union_3; assumption). -unfold V_pref_reg_mreg. -unfold AE_reg_mreg in H0. -rewrite SetRegMreg.fold_1 in *. -induction (SetRegMreg.elements (pref_reg_mreg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.mreg_to_Reg (snd a), Some N0) e) in H1. -rewrite RegFacts.fold_left_assoc. -destruct (eq_charac _ _ H1); destruct H2 as [H2 H3]. -change_rewrite. split. -apply VertexSet.add_1. assumption. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -change_rewrite. split. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -apply VertexSet.add_1. assumption. -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.mreg_to_Reg (snd y)) s) - (Regs.mreg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.mreg_to_Reg (snd z)) - (Regs.mreg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.mreg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.mreg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -rewrite RegFacts.fold_left_assoc. -split; do 2 apply VertexSet.add_2. -apply (proj1 (IHl H1)). -apply (proj2 (IHl H1)). -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.mreg_to_Reg (snd y)) s) - (Regs.mreg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.mreg_to_Reg (snd z)) - (Regs.mreg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.mreg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.mreg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. - -unfold interfgraph_interference_edges in H. -destruct (EdgeSet.union_1 H). -cut (VertexSet.In (fst_ext e) (V_interf_reg_reg g) /\ - VertexSet.In (snd_ext e) (V_interf_reg_reg g)). -intro H1. destruct H1 as [H1 H2]. split; -(apply VertexSet.union_2; assumption). -unfold V_interf_reg_reg. -unfold IE_reg_reg in H0. -rewrite SetRegReg.fold_1 in *. -induction (SetRegReg.elements (interf_reg_reg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.reg_to_Reg (snd a), Some N0) e) in H1. -rewrite RegFacts.fold_left_assoc. -destruct (eq_charac _ _ H1); destruct H2 as [H2 H3]. -change_rewrite. split. -apply VertexSet.add_1. assumption. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -change_rewrite. split. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -apply VertexSet.add_1. assumption. -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.reg_to_Reg (snd y)) s) - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.reg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.reg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -rewrite RegFacts.fold_left_assoc. -split; do 2 apply VertexSet.add_2. -apply (proj1 (IHl H1)). -apply (proj2 (IHl H1)). -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.reg_to_Reg (snd y)) s) - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.reg_to_Reg (snd z)) - (Regs.reg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.reg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.reg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. - -cut (VertexSet.In (fst_ext e) (V_interf_reg_mreg g) /\ - VertexSet.In (snd_ext e) (V_interf_reg_mreg g)). -intro H1. destruct H1 as [H1 H2]. split; -(apply VertexSet.union_3; apply VertexSet.union_2; assumption). -unfold V_interf_reg_mreg. -unfold IE_reg_mreg in H0. -rewrite SetRegMreg.fold_1 in *. -induction (SetRegMreg.elements (interf_reg_mreg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.mreg_to_Reg (snd a), Some N0) e) in H1. -rewrite RegFacts.fold_left_assoc. -destruct (eq_charac _ _ H1); destruct H2 as [H2 H3]. -change_rewrite. split. -apply VertexSet.add_1. assumption. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -change_rewrite. split. -apply VertexSet.add_2. apply VertexSet.add_1. assumption. -apply VertexSet.add_1. assumption. -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.mreg_to_Reg (snd y)) s) - (Regs.mreg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.mreg_to_Reg (snd z)) - (Regs.mreg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.mreg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.mreg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -rewrite RegFacts.fold_left_assoc. -split; do 2 apply VertexSet.add_2. -apply (proj1 (IHl H1)). -apply (proj2 (IHl H1)). -intros y z s. -rewrite (Props.add_add (VertexSet.add (Regs.mreg_to_Reg (snd y)) s) - (Regs.mreg_to_Reg (snd z)) - (Regs.reg_to_Reg (fst y))). -rewrite (Props.add_add s - (Regs.mreg_to_Reg (snd z)) - (Regs.mreg_to_Reg (snd y))). -set (s' := VertexSet.add (Regs.mreg_to_Reg (snd z) ) s). -rewrite (Props.add_add s' - (Regs.mreg_to_Reg (snd y)) - (Regs.reg_to_Reg (fst z))). -apply Props.add_add. -intros. apply Props.Dec.F.add_m. -apply Regs.eq_refl. -apply Props.Dec.F.add_m;[apply Regs.eq_refl|assumption]. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. -Qed. -*) - -Lemma vertices_empty_map : forall x, -VertexMap.In x empty_map <-> -VertexSet.In x interfgraph_vertices. - -Proof. -unfold empty_map. split; intros. -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x VertexSet.empty m)) in *. -rewrite VertexSet.fold_1 in *. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH interfgraph_vertices). clear HH. intro HH. -induction (VertexSet.elements interfgraph_vertices). simpl in H. -rewrite MapFacts.empty_in_iff in H. inversion H. -cut (EqualSetMap (fold_left f' (a :: l) (VertexMap.empty VertexSet.t)) - (f' (fold_left f' l (VertexMap.empty VertexSet.t)) a)). intro. -generalize (H0 x). clear H0. intro H0. inversion H0. subst. -simpl in H. rewrite MapFacts.in_find_iff in H. rewrite <-H2 in H. -elim H. auto. -set (tmp := fold_left f' l (VertexMap.empty VertexSet.t)) in *. -unfold f', f in H2. -destruct (Vertex.eq_dec x a). -rewrite e. apply HH. left. apply Regs.eq_refl. -apply IHl. -rewrite MapFacts.add_neq_o in H2. -rewrite MapFacts.in_find_iff. rewrite <-H2. congruence. -auto. - -intros. apply HH. right. auto. - -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f', f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -auto. -auto. - -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x VertexSet.empty m)) in *. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH _ _ H). clear HH. intro HH. -induction (VertexSet.elements interfgraph_vertices). -inversion HH. -cut (EqualSetMap (fold_left f' (a :: l) (VertexMap.empty VertexSet.t)) - (f' (fold_left f' l (VertexMap.empty VertexSet.t)) a)). intro. -generalize (H0 x). clear H0. intro H0. inversion H0. subst. -set (tmp := fold_left f' l (VertexMap.empty VertexSet.t)) in *. -unfold f', f in H3. -destruct (Vertex.eq_dec x a). -simpl. rewrite MapFacts.in_find_iff. -rewrite MapFacts.add_eq_o in H3. congruence. -apply Regs.eq_sym. auto. -inversion HH; subst. -elim (n H4). -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff in IHl. rewrite <-H3 in IHl. -elim (IHl H4 (refl_equal _)). -auto. -simpl. rewrite MapFacts.in_find_iff. rewrite <-H1. congruence. - -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f', f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -auto. -auto. -Qed. - - -Lemma extremities_imap : forall x, -VertexMap.In x interf_map <-> -VertexSet.In x interfgraph_vertices. - -Proof. -split; intros. -unfold interf_map in H. -unfold set2map in H. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) - (new_adj_set (snd_ext e) (fst_ext e) m) m))) in *. -rewrite EdgeSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -cut (forall e, EdgeSet.In e interfgraph_interference_edges -> - VertexSet.In (fst_ext e) interfgraph_vertices - /\ VertexSet.In (snd_ext e) interfgraph_vertices). intro. - -generalize EdgeSet.elements_2. intro HH. -generalize (HH interfgraph_interference_edges). clear HH. intro HH. - -induction (EdgeSet.elements interfgraph_interference_edges). simpl in H. -apply (proj1 (vertices_empty_map x)). auto. - -cut (EqualSetMap (fold_left f' (a :: l) empty_map) - (f' (fold_left f' l empty_map ) a)). intro. -generalize (H1 x). clear H1. intro H1. inversion H1. subst. -simpl in H. rewrite MapFacts.in_find_iff in H. rewrite <-H3 in H. -elim H. auto. - -set (tmp := fold_left f' l empty_map) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite e. apply (H0 a). -apply HH. left. apply E.eq_refl. -destruct (Regs.eq_dec x (snd_ext a)). -rewrite e. apply (H0 a). -apply HH. left. apply E.eq_refl. -apply IHl. -do 2 rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff. rewrite <-H3. congruence. -auto. -auto. -auto. - -intros. apply HH. right. auto. - -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -intro. elim n. rewrite e. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite e. auto. -intro. elim n. rewrite e. auto. -apply (Regs.eq_sym e). -intro. elim n0. auto. -intro. elim n. auto. -apply (Regs.eq_sym e). -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Regs.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H1. auto. -intro. elim n0. rewrite H1. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -constructor. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold f'. unfold f. unfold EqualSetMap. intros. -destruct (Regs.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H1 (fst_ext a0)). intro. -inversion H2. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H1 (snd_ext a0)). intro. -inversion H2. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H2. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -intro. elim n0. rewrite H2. auto. -intro. elim n. rewrite H2. auto. -auto. -auto. - -intros. apply extremities_interf_graph. right. auto. - -rewrite <-vertices_empty_map in H. -unfold interf_map. unfold set2map. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) (new_adj_set (snd_ext e) (fst_ext e) m) m))) in *. -rewrite EdgeSet.fold_1. -set (f' := fun e a => f a e) in *. -induction (EdgeSet.elements interfgraph_interference_edges). simpl. auto. -cut (EqualSetMap (fold_left f' (a :: l) empty_map) - (f' (fold_left f' l empty_map) a)). intro. -generalize (H0 x). clear H0. intro H0. inversion H0. subst. -set (tmp := fold_left f' l empty_map) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H3. congruence. -apply (Regs.eq_sym e). -destruct (Vertex.eq_dec x (snd_ext a)). -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_eq_o in H3. -congruence. -apply (Regs.eq_sym e). -auto. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff in IHl. rewrite <-H3 in IHl. -congruence. -auto. -auto. - -simpl. rewrite MapFacts.in_find_iff. rewrite <-H1. congruence. -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -intro. elim n. rewrite e. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite e. auto. -intro. elim n. rewrite e. auto. -apply (Regs.eq_sym e). -intro. elim n0. auto. -intro. elim n. auto. -apply (Regs.eq_sym e). -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Regs.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H0. auto. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -constructor. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold f'. unfold f. unfold EqualSetMap. intros. -destruct (Regs.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (fst_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (snd_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -auto. -auto. -Qed. - -Lemma extremities_pmap : forall x, -VertexMap.In x pref_map <-> -VertexSet.In x interfgraph_vertices. - -Proof. -split; intros. -unfold pref_map in H. -unfold set2map in H. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) - (new_adj_set (snd_ext e) (fst_ext e) m) m))) in *. -rewrite EdgeSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -cut (forall e, EdgeSet.In e interfgraph_affinity_edges -> - VertexSet.In (fst_ext e) interfgraph_vertices - /\ VertexSet.In (snd_ext e) interfgraph_vertices). intro. - -generalize EdgeSet.elements_2. intro HH. -generalize (HH interfgraph_affinity_edges). clear HH. intro HH. - -induction (EdgeSet.elements interfgraph_affinity_edges). simpl in H. -apply (proj1 (vertices_empty_map x)). auto. - -cut (EqualSetMap (fold_left f' (a :: l) empty_map) - (f' (fold_left f' l empty_map) a)). intro. -generalize (H1 x). clear H1. intro H1. inversion H1. subst. -simpl in H. rewrite MapFacts.in_find_iff in H. rewrite <-H3 in H. -elim H. auto. - -set (tmp := fold_left f' l empty_map) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite e. apply (H0 a). -apply HH. left. apply E.eq_refl. -destruct (Regs.eq_dec x (snd_ext a)). -rewrite e. apply (H0 a). -apply HH. left. apply E.eq_refl. -apply IHl. -do 2 rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff. rewrite <-H3. congruence. -auto. -auto. -auto. - -intros. apply HH. right. auto. - -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -intro. elim n. rewrite e. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite e. auto. -intro. elim n. rewrite e. auto. -apply (Regs.eq_sym e). -intro. elim n0. auto. -intro. elim n. auto. -apply (Regs.eq_sym e). -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Regs.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H1. auto. -intro. elim n0. rewrite H1. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -constructor. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold f'. unfold f. unfold EqualSetMap. intros. -destruct (Regs.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H1 (fst_ext a0)). intro. -inversion H2. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H1 (snd_ext a0)). intro. -inversion H2. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H2. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -intro. elim n0. rewrite H2. auto. -intro. elim n. rewrite H2. auto. -auto. -auto. - -intros. apply extremities_interf_graph. left. auto. - -rewrite <-vertices_empty_map in H. -unfold pref_map. unfold set2map. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) (new_adj_set (snd_ext e) (fst_ext e) m) m))) in *. -rewrite EdgeSet.fold_1. -set (f' := fun e a => f a e) in *. -induction (EdgeSet.elements interfgraph_affinity_edges). simpl. auto. -cut (EqualSetMap (fold_left f' (a :: l) empty_map) - (f' (fold_left f' l empty_map) a)). intro. -generalize (H0 x). clear H0. intro H0. inversion H0. subst. -set (tmp := fold_left f' l empty_map) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H3. congruence. -apply (Regs.eq_sym e). -destruct (Vertex.eq_dec x (snd_ext a)). -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_eq_o in H3. -congruence. -apply (Regs.eq_sym e). -auto. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff in IHl. rewrite <-H3 in IHl. -congruence. -auto. -auto. - -simpl. rewrite MapFacts.in_find_iff. rewrite <-H1. congruence. -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -intro. elim n. rewrite e. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite e. auto. -intro. elim n. rewrite e. auto. -apply (Regs.eq_sym e). -intro. elim n0. auto. -intro. elim n. auto. -apply (Regs.eq_sym e). -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Regs.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H0. auto. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -constructor. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold f'. unfold f. unfold EqualSetMap. intros. -destruct (Regs.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (fst_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (snd_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -auto. -auto. -Qed. - -Lemma set2map_charac : forall x y s m, -VertexSet.In x (adj_set y (set2map s m)) -> -(exists w, EdgeSet.In (x,y,w) s) \/ VertexSet.In x (adj_set y m). - -Proof. -intros. -unfold set2map in H. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) - (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) - (new_adj_set (snd_ext e) (fst_ext e) m) m))) in *. -rewrite EdgeSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize EdgeSet.elements_2. intro HH. -generalize (HH s). clear HH. intro HH. -induction (EdgeSet.elements s). simpl in H. right. auto. -cut (EqualSetMap (fold_left f' (a :: l) m) - (f' (fold_left f' l m) a)). intro. -generalize (H0 y). clear H0. intro H0. inversion H0. subst. -set (tmp := fold_left f' l m) in *. -unfold f', f in H3. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H3. -congruence. -apply Regs.eq_sym. auto. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_eq_o in H3. -congruence. -apply Regs.eq_sym. auto. -auto. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_neq_o in H3. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -elim (VertexSet.empty_1 H). -auto. -auto. -simpl in H. unfold adj_set in H. rewrite <-H1 in H. -rewrite H3 in H. -set (tmp := fold_left f' l m) in *. -unfold f', f in H2. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H2. inversion H2. -rewrite H5 in H. -unfold new_adj_set in H. -case_eq (VertexMap.find (fst_ext a) tmp); intros. -rewrite H4 in H. -destruct (Regs.eq_dec (snd_ext a) x). -left. exists (get_weight a). -apply HH. left. -fold (eq (x,y,get_weight a) a). -rewrite edge_comm. Eq_eq. -generalize (VertexSet.add_3 n H). intro. -apply IHl. -unfold adj_set. rewrite (MapFacts.find_o _ e). -rewrite H4. auto. -intros. apply HH. right. auto. -rewrite H4 in H. -left. exists (get_weight a). -apply HH. left. -fold (eq (x,y,get_weight a) a). -rewrite edge_comm. Eq_eq. -apply Regs.eq_sym. apply VertexSet.singleton_1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set in H. -case_eq (VertexMap.find (snd_ext a) tmp); intros. -rewrite H2 in H. -destruct (Vertex.eq_dec (fst_ext a) x). -left. exists (get_weight a). -apply HH. left. -fold (eq (x,y,get_weight a) a). Eq_eq. -apply IHl. -generalize (VertexSet.add_3 n0 H). intro. -unfold new_adj_set in H3. rewrite H2 in H3. -unfold adj_set. rewrite (MapFacts.find_o _ e). rewrite H2. -auto. -intros. apply HH. right. auto. -rewrite H2 in H. -left. exists (get_weight a). -apply HH. left. -fold (eq (x,y,get_weight a) a). Eq_eq. -apply Regs.eq_sym. apply VertexSet.singleton_1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -apply IHl. -unfold adj_set. rewrite <-H2. auto. -intros. apply HH. right. auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold f', f, EqualSetMap. intros. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H0. auto. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold EqualSetMap, f', f. intros. -destruct (Vertex.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (fst_ext a0)). intro. -inversion H1. apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. -apply Regs.eq_refl. -auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (snd_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. -apply Regs.eq_refl. -auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -auto. -auto. -auto. -auto. -Qed. - -Lemma adj_set_empty : forall x, -VertexSet.Equal (adj_set x empty_map) VertexSet.empty. - -Proof. -intros. -unfold empty_map. -set (f := (fun (x0 : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x0 VertexSet.empty m)) in *. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements interfgraph_vertices). -simpl. unfold adj_set. simpl. reflexivity. -cut (EqualSetMap (fold_left f' (a :: l) (VertexMap.empty VertexSet.t)) - (f' (fold_left f' l (VertexMap.empty VertexSet.t)) a)). intro. -generalize (H x). clear H. intro H. inversion H. subst. -set (tmp := fold_left f' l (VertexMap.empty VertexSet.t)) in *. -unfold f', f in H2. -destruct (Regs.eq_dec x a). rewrite MapFacts.add_eq_o in H2. congruence. -apply Regs.eq_sym. auto. -simpl. unfold adj_set. rewrite <-H1. apply VertexSet.eq_refl. -set (tmp := fold_left f' l (VertexMap.empty VertexSet.t)) in *. -unfold f', f in H1. -destruct (Regs.eq_dec x a). rewrite MapFacts.add_eq_o in H1. -unfold adj_set. simpl. rewrite <-H0. inversion H1. -rewrite H4 in H2. auto. -apply Regs.eq_sym. auto. -simpl. unfold adj_set. rewrite <-H0. -rewrite MapFacts.add_neq_o in H1. -unfold adj_set in IHl. rewrite <-H1 in IHl. -rewrite H2. auto. -auto. - -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f', f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -auto. -auto. -Qed. - -Lemma affinity_weights_interf_graph : -forall e, EdgeSet.In e interfgraph_affinity_edges -> -get_weight e = Some N0. - -Proof. -intros e H. -unfold interfgraph_affinity_edges in H. -generalize (proj1 (resolve_conflicts_2 _ _ _ H)). clear H. intro H. -destruct (EdgeSet.union_1 H). -unfold AE_reg_reg in H0. -rewrite SetRegReg.fold_1 in H0. -induction (SetRegReg.elements (pref_reg_reg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.reg_to_Reg (snd a), Some N0) e) in H1. -rewrite <-(get_weight_m _ _ H1). auto. -apply IHl. assumption. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. -unfold AE_reg_mreg in H0. -rewrite SetRegMreg.fold_1 in H0. -induction (SetRegMreg.elements (pref_reg_mreg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.mreg_to_Reg (snd a), Some N0) e) in H1. -rewrite <-(get_weight_m _ _ H1). auto. -apply IHl. assumption. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. -Qed. - -Lemma interference_weights_interf_graph : -forall e, EdgeSet.In e interfgraph_interference_edges -> -get_weight e = None. - -Proof. -intros e H. -unfold interfgraph_affinity_edges in H. -destruct (EdgeSet.union_1 H). -unfold IE_reg_reg in H0. -rewrite SetRegReg.fold_1 in H0. -induction (SetRegReg.elements (interf_reg_reg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.reg_to_Reg (snd a), Some N0) e) in H1. -rewrite <-(get_weight_m _ _ H1). auto. -apply IHl. assumption. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. -unfold IE_reg_mreg in H0. -rewrite SetRegMreg.fold_1 in H0. -induction (SetRegMreg.elements (interf_reg_mreg g)). -simpl in H0. elim (EdgeSet.empty_1 H0). -rewrite MEdgeFacts.fold_left_assoc in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (Regs.reg_to_Reg (fst a), Regs.mreg_to_Reg (snd a), Some N0) e) in H1. -rewrite <-(get_weight_m _ _ H1). auto. -apply IHl. assumption. -intros;apply RegRegProps.add_add. -intros;apply RegRegProps.Dec.F.add_m;[apply eq_refl|assumption]. -Qed. - -Lemma simple_graph : forall x y, - VertexSet.In x (adj_set y interf_map ) /\ - VertexSet.In x (adj_set y pref_map ) -> False. - -Proof. -intros. destruct H. -unfold interf_map in H. -generalize (set2map_charac x y interfgraph_interference_edges empty_map H). -clear H. intro H. destruct H. destruct H. -unfold pref_map in H0. -generalize (set2map_charac x y interfgraph_affinity_edges empty_map H0). -clear H0. intro H0. destruct H0. destruct H0. -generalize (resolve_conflicts_2 (x,y,x1) _ _ H0). intro. -destruct H1. elim H2. -change_rewrite. -generalize (interference_weights_interf_graph (x,y,x0) H). simpl. intro. -rewrite H3 in H. assumption. -rewrite adj_set_empty in H0. elim (VertexSet.empty_1 H0). -rewrite adj_set_empty in H. elim (VertexSet.empty_1 H). -Qed. - -Lemma sym_imap : forall x y, - VertexSet.In x (adj_set y interf_map ) -> - VertexSet.In y (adj_set x interf_map). - -Proof. -intros. -unfold interf_map, set2map in *. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) (new_adj_set (snd_ext e) (fst_ext e) m) - m))) in *. -rewrite EdgeSet.fold_1. rewrite EdgeSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -induction (EdgeSet.elements interfgraph_interference_edges). -simpl in H. -rewrite adj_set_empty in H. elim (VertexSet.empty_1 H). -cut (EqualSetMap (fold_left f' (a :: l) empty_map) - (f' (fold_left f' l empty_map) a)). intro. -generalize (H0 y). intro. inversion H1. simpl in *. -unfold adj_set in H. rewrite <-H3 in H. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l empty_map) in *. -unfold f', f in H3. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H3. -unfold adj_set in H. simpl in H. rewrite <-H2 in H. -inversion H3. subst. clear H3. -rewrite H4 in H. -unfold new_adj_set in H. -generalize (H0 x). intro. inversion H3. subst. -unfold f', f in H7. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H7. congruence. -apply Regs.eq_sym. auto. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_neq_o in H7. rewrite MapFacts.add_eq_o in H7. -congruence. -apply Regs.eq_sym. auto. -auto. -rewrite MapFacts.add_neq_o in H7. -rewrite MapFacts.add_neq_o in H7. -case_eq (VertexMap.find (fst_ext a) tmp); intros; rewrite H5 in H. -generalize (VertexSet.add_3 n0 H). clear H. intro H. -assert (VertexSet.In y (adj_set x tmp)). -apply IHl. unfold adj_set. rewrite (MapFacts.find_o _ e). -rewrite H5. auto. -unfold adj_set in H8. rewrite <-H7 in H8. elim (VertexSet.empty_1 H8). -elim (n0 (VertexSet.singleton_1 H)). -auto. -auto. - -simpl. unfold adj_set. rewrite <-H5. -rewrite H7. -unfold f', f in H6. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H6. -inversion H6. subst. clear H6. -unfold new_adj_set. case_eq (VertexMap.find (fst_ext a) tmp); intros. -rewrite H6 in H. -destruct (Vertex.eq_dec y (snd_ext a)). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.add_2. -assert (VertexSet.In y (adj_set x tmp)). -apply IHl. unfold adj_set. -rewrite (MapFacts.find_o _ e). -rewrite H6. -destruct (Vertex.eq_dec (snd_ext a) x). -elim n. rewrite e. rewrite <-e0. apply Regs.eq_sym. auto. -apply (VertexSet.add_3 n0 H). -unfold adj_set in H8. -rewrite (MapFacts.find_o _ e0) in H8. rewrite H6 in H8. auto. -rewrite H6 in H. -rewrite e. rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_eq_o in H6. -inversion H6. subst. clear H6. -unfold new_adj_set. -destruct (VertexMap.find (snd_ext a) tmp). -apply VertexSet.add_1. -apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -clear H0 H1. -assert (VertexSet.In y (adj_set x tmp)). -apply IHl. -unfold adj_set. rewrite (MapFacts.find_o _ e). -case_eq (VertexMap.find (fst_ext a) tmp); intros. -rewrite H0 in H. -apply (VertexSet.add_3 n0 H). -rewrite H0 in H. elim (n0 (VertexSet.singleton_1 H)). -unfold adj_set in H0. rewrite <-H6 in H0. auto. -auto. -auto. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H3. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_eq_o in H3. -unfold adj_set in H. simpl in H. rewrite <-H2 in H. -rewrite H4 in H. -inversion H3. subst. clear H3. -unfold new_adj_set in H. -case_eq (VertexMap.find (snd_ext a) tmp); intros. -rewrite H3 in H. -destruct (Vertex.eq_dec (fst_ext a) x). -unfold adj_set. simpl. -generalize (H0 x). clear H0. intro. inversion H0. -unfold f',f in H7. -rewrite MapFacts.add_eq_o in H7. -congruence. -apply Regs.eq_sym. auto. -unfold f', f in H6. -rewrite MapFacts.add_eq_o in H6. inversion H6. subst. clear H6. -rewrite H7. -unfold new_adj_set in H4. rewrite H3 in H4. -unfold new_adj_set. -destruct (VertexMap.find (fst_ext a) tmp). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -unfold adj_set. simpl. -generalize (H0 x). clear H0. intro. inversion H0. -unfold f',f in H7. -rewrite MapFacts.add_neq_o in H7. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_eq_o in H7. congruence. auto. -rewrite MapFacts.add_neq_o in H7. -unfold adj_set in IHl. rewrite <-H7 in IHl. apply IHl. -rewrite (MapFacts.find_o _ e). rewrite H3. -apply (VertexSet.add_3 n0 H). -auto. -auto. -unfold f', f in H6. -rewrite MapFacts.add_neq_o in H6. -destruct (Vertex.eq_dec x (snd_ext a)). -rewrite MapFacts.add_eq_o in H6. inversion H6. subst. clear H6. -rewrite H7. -unfold new_adj_set. rewrite H3. -apply VertexSet.add_2. -unfold adj_set in IHl. rewrite <-(MapFacts.find_o _ e0) in H3. -rewrite H3 in IHl. apply IHl. -rewrite (MapFacts.find_o _ e). rewrite <-(MapFacts.find_o _ e0). -rewrite H3. -apply (VertexSet.add_3 n0 H). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -rewrite H7. -unfold adj_set in IHl. rewrite <-H6 in IHl. -apply IHl. rewrite (MapFacts.find_o _ e). -rewrite H3. -apply (VertexSet.add_3 n0 H). -auto. -auto. -rewrite H3 in H. -generalize (H0 x). intro. inversion H5. subst. -unfold f',f in H8. -rewrite MapFacts.add_eq_o in H8. -congruence. -apply VertexSet.singleton_1. auto. -unfold f', f in H7. -rewrite MapFacts.add_eq_o in H7. -inversion H7. subst. clear H7. -simpl. unfold adj_set. rewrite <-H6. -rewrite H8. unfold new_adj_set. -destruct (VertexMap.find (fst_ext a) tmp). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply (VertexSet.singleton_1 H). -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H3. -generalize (H0 x). intro. inversion H5. subst. -unfold f', f in H8. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H8. -congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H8. -destruct (Vertex.eq_dec x (snd_ext a)). -rewrite MapFacts.add_eq_o in H8. -congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H8. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -unfold adj_set in IHl. -assert (VertexSet.In y VertexSet.empty). -rewrite <-H8 in IHl. apply IHl. -rewrite <-H3. rewrite <-H4. auto. -elim (VertexSet.empty_1 H6). -auto. -auto. -unfold f',f in H7. -destruct (Vertex.eq_dec (fst_ext a) x). -rewrite MapFacts.add_eq_o in H7. -inversion H7. subst. clear H7. -unfold adj_set. simpl. rewrite <-H6. -rewrite H8. -unfold new_adj_set. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -case_eq (VertexMap.find (fst_ext a) tmp); intros. -unfold adj_set in IHl. -rewrite <-(MapFacts.find_o _ e) in IHl. -rewrite H7 in IHl. -apply VertexSet.add_2. apply IHl. -rewrite <-H3. rewrite <-H4. auto. -assert (VertexSet.In y VertexSet.empty). -unfold adj_set in IHl. rewrite (MapFacts.find_o _ e) in H7. -rewrite H7 in IHl. apply IHl. -rewrite <-H3. rewrite <-H4. auto. -elim (VertexSet.empty_1 H9). -auto. -rewrite MapFacts.add_neq_o in H7. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_eq_o in H7. -inversion H7. subst. clear H7. -simpl. unfold adj_set. rewrite <-H6. -rewrite H8. unfold new_adj_set. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -rewrite (MapFacts.find_o _ e). -unfold adj_set in IHl. -case_eq (VertexMap.find x tmp); intros. -rewrite H7 in IHl. -apply VertexSet.add_2. -apply IHl. -rewrite <-H3. rewrite <-H4. auto. -assert (VertexSet.In y VertexSet.empty). -unfold adj_set in IHl. rewrite H7 in IHl. -apply IHl. rewrite <-H3. rewrite <-H4. auto. -elim (VertexSet.empty_1 H9). -auto. -rewrite MapFacts.add_neq_o in H7. -simpl. unfold adj_set. rewrite <-H6. -rewrite H8. -unfold adj_set in IHl. rewrite <-H7 in IHl. rewrite <-H3 in IHl. -apply IHl. rewrite <-H4. simpl in H. unfold adj_set in H. rewrite <-H2 in H. auto. -auto. -auto. -auto. -auto. - -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -intro. elim n. rewrite e. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite e. auto. -intro. elim n. rewrite e. auto. -apply (Regs.eq_sym e). -intro. elim n0. auto. -intro. elim n. auto. -apply (Regs.eq_sym e). -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Regs.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H0. auto. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -constructor. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold f'. unfold f. unfold EqualSetMap. intros. -destruct (Regs.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (fst_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (snd_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -auto. -auto. -Qed. - -Lemma sym_pmap : forall x y, - VertexSet.In x (adj_set y pref_map) -> - VertexSet.In y (adj_set x pref_map). - -Proof. -intros. -unfold pref_map, set2map in *. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) (new_adj_set (snd_ext e) (fst_ext e) m) - m))) in *. -rewrite EdgeSet.fold_1. rewrite EdgeSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -induction (EdgeSet.elements interfgraph_affinity_edges). -simpl in H. -rewrite adj_set_empty in H. elim (VertexSet.empty_1 H). -cut (EqualSetMap (fold_left f' (a :: l) empty_map) - (f' (fold_left f' l empty_map) a)). intro. -generalize (H0 y). intro. inversion H1. simpl in *. -unfold adj_set in H. rewrite <-H3 in H. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l empty_map) in *. -unfold f', f in H3. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H3. -unfold adj_set in H. simpl in H. rewrite <-H2 in H. -inversion H3. subst. clear H3. -rewrite H4 in H. -unfold new_adj_set in H. -generalize (H0 x). intro. inversion H3. subst. -unfold f', f in H7. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H7. congruence. -apply Regs.eq_sym. auto. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_neq_o in H7. rewrite MapFacts.add_eq_o in H7. -congruence. -apply Regs.eq_sym. auto. -auto. -rewrite MapFacts.add_neq_o in H7. -rewrite MapFacts.add_neq_o in H7. -case_eq (VertexMap.find (fst_ext a) tmp); intros; rewrite H5 in H. -generalize (VertexSet.add_3 n0 H). clear H. intro H. -assert (VertexSet.In y (adj_set x tmp)). -apply IHl. unfold adj_set. rewrite (MapFacts.find_o _ e). -rewrite H5. auto. -unfold adj_set in H8. rewrite <-H7 in H8. elim (VertexSet.empty_1 H8). -elim (n0 (VertexSet.singleton_1 H)). -auto. -auto. - -simpl. unfold adj_set. rewrite <-H5. -rewrite H7. -unfold f', f in H6. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H6. -inversion H6. subst. clear H6. -unfold new_adj_set. case_eq (VertexMap.find (fst_ext a) tmp); intros. -rewrite H6 in H. -destruct (Vertex.eq_dec y (snd_ext a)). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.add_2. -assert (VertexSet.In y (adj_set x tmp)). -apply IHl. unfold adj_set. -rewrite (MapFacts.find_o _ e). -rewrite H6. -destruct (Vertex.eq_dec (snd_ext a) x). -elim n. rewrite e. rewrite <-e0. apply Regs.eq_sym. auto. -apply (VertexSet.add_3 n0 H). -unfold adj_set in H8. -rewrite (MapFacts.find_o _ e0) in H8. rewrite H6 in H8. auto. -rewrite H6 in H. -rewrite e. rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_eq_o in H6. -inversion H6. subst. clear H6. -unfold new_adj_set. -destruct (VertexMap.find (snd_ext a) tmp). -apply VertexSet.add_1. -apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -clear H0 H1. -assert (VertexSet.In y (adj_set x tmp)). -apply IHl. -unfold adj_set. rewrite (MapFacts.find_o _ e). -case_eq (VertexMap.find (fst_ext a) tmp); intros. -rewrite H0 in H. -apply (VertexSet.add_3 n0 H). -rewrite H0 in H. elim (n0 (VertexSet.singleton_1 H)). -unfold adj_set in H0. rewrite <-H6 in H0. auto. -auto. -auto. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H3. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_eq_o in H3. -unfold adj_set in H. simpl in H. rewrite <-H2 in H. -rewrite H4 in H. -inversion H3. subst. clear H3. -unfold new_adj_set in H. -case_eq (VertexMap.find (snd_ext a) tmp); intros. -rewrite H3 in H. -destruct (Vertex.eq_dec (fst_ext a) x). -unfold adj_set. simpl. -generalize (H0 x). clear H0. intro. inversion H0. -unfold f',f in H7. -rewrite MapFacts.add_eq_o in H7. -congruence. -apply Regs.eq_sym. auto. -unfold f', f in H6. -rewrite MapFacts.add_eq_o in H6. inversion H6. subst. clear H6. -rewrite H7. -unfold new_adj_set in H4. rewrite H3 in H4. -unfold new_adj_set. -destruct (VertexMap.find (fst_ext a) tmp). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -unfold adj_set. simpl. -generalize (H0 x). clear H0. intro. inversion H0. -unfold f',f in H7. -rewrite MapFacts.add_neq_o in H7. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_eq_o in H7. congruence. auto. -rewrite MapFacts.add_neq_o in H7. -unfold adj_set in IHl. rewrite <-H7 in IHl. apply IHl. -rewrite (MapFacts.find_o _ e). rewrite H3. -apply (VertexSet.add_3 n0 H). -auto. -auto. -unfold f', f in H6. -rewrite MapFacts.add_neq_o in H6. -destruct (Vertex.eq_dec x (snd_ext a)). -rewrite MapFacts.add_eq_o in H6. inversion H6. subst. clear H6. -rewrite H7. -unfold new_adj_set. rewrite H3. -apply VertexSet.add_2. -unfold adj_set in IHl. rewrite <-(MapFacts.find_o _ e0) in H3. -rewrite H3 in IHl. apply IHl. -rewrite (MapFacts.find_o _ e). rewrite <-(MapFacts.find_o _ e0). -rewrite H3. -apply (VertexSet.add_3 n0 H). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -rewrite H7. -unfold adj_set in IHl. rewrite <-H6 in IHl. -apply IHl. rewrite (MapFacts.find_o _ e). -rewrite H3. -apply (VertexSet.add_3 n0 H). -auto. -auto. -rewrite H3 in H. -generalize (H0 x). intro. inversion H5. subst. -unfold f',f in H8. -rewrite MapFacts.add_eq_o in H8. -congruence. -apply VertexSet.singleton_1. auto. -unfold f', f in H7. -rewrite MapFacts.add_eq_o in H7. -inversion H7. subst. clear H7. -simpl. unfold adj_set. rewrite <-H6. -rewrite H8. unfold new_adj_set. -destruct (VertexMap.find (fst_ext a) tmp). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply (VertexSet.singleton_1 H). -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H3. -generalize (H0 x). intro. inversion H5. subst. -unfold f', f in H8. -destruct (Vertex.eq_dec x (fst_ext a)). -rewrite MapFacts.add_eq_o in H8. -congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H8. -destruct (Vertex.eq_dec x (snd_ext a)). -rewrite MapFacts.add_eq_o in H8. -congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H8. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -unfold adj_set in IHl. -assert (VertexSet.In y VertexSet.empty). -rewrite <-H8 in IHl. apply IHl. -rewrite <-H3. rewrite <-H4. auto. -elim (VertexSet.empty_1 H6). -auto. -auto. -unfold f',f in H7. -destruct (Vertex.eq_dec (fst_ext a) x). -rewrite MapFacts.add_eq_o in H7. -inversion H7. subst. clear H7. -unfold adj_set. simpl. rewrite <-H6. -rewrite H8. -unfold new_adj_set. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -case_eq (VertexMap.find (fst_ext a) tmp); intros. -unfold adj_set in IHl. -rewrite <-(MapFacts.find_o _ e) in IHl. -rewrite H7 in IHl. -apply VertexSet.add_2. apply IHl. -rewrite <-H3. rewrite <-H4. auto. -assert (VertexSet.In y VertexSet.empty). -unfold adj_set in IHl. rewrite (MapFacts.find_o _ e) in H7. -rewrite H7 in IHl. apply IHl. -rewrite <-H3. rewrite <-H4. auto. -elim (VertexSet.empty_1 H9). -auto. -rewrite MapFacts.add_neq_o in H7. -destruct (Vertex.eq_dec (snd_ext a) x). -rewrite MapFacts.add_eq_o in H7. -inversion H7. subst. clear H7. -simpl. unfold adj_set. rewrite <-H6. -rewrite H8. unfold new_adj_set. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -rewrite (MapFacts.find_o _ e). -unfold adj_set in IHl. -case_eq (VertexMap.find x tmp); intros. -rewrite H7 in IHl. -apply VertexSet.add_2. -apply IHl. -rewrite <-H3. rewrite <-H4. auto. -assert (VertexSet.In y VertexSet.empty). -unfold adj_set in IHl. rewrite H7 in IHl. -apply IHl. rewrite <-H3. rewrite <-H4. auto. -elim (VertexSet.empty_1 H9). -auto. -rewrite MapFacts.add_neq_o in H7. -simpl. unfold adj_set. rewrite <-H6. -rewrite H8. -unfold adj_set in IHl. rewrite <-H7 in IHl. rewrite <-H3 in IHl. -apply IHl. rewrite <-H4. simpl in H. unfold adj_set in H. rewrite <-H2 in H. auto. -auto. -auto. -auto. -auto. - -apply fold_left_assoc_map. -intros. -unfold f'. unfold f. -unfold EqualSetMap. intro. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -intro. elim n. rewrite e. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite e. auto. -intro. elim n. rewrite e. auto. -apply (Regs.eq_sym e). -intro. elim n0. auto. -intro. elim n. auto. -apply (Regs.eq_sym e). -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Regs.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite (MapFacts.find_o _ (Regs.eq_sym e)). -rewrite (MapFacts.find_o _ (Regs.eq_sym e0)). -destruct (VertexMap.find x0 s). -apply Props.add_add. -do 2 rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. auto. -rewrite <-e0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H0. auto. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -constructor. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s). -constructor. apply VertexSet.eq_refl. -constructor. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold f'. unfold f. unfold EqualSetMap. intros. -destruct (Regs.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (fst_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Regs.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (snd_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. apply Regs.eq_refl. auto. -apply Regs.eq_sym. auto. -intro. elim n. rewrite H1. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -intro. elim n0. rewrite H1. auto. -intro. elim n. rewrite H1. auto. -auto. -auto. -Qed. - -Lemma set_reg_reg_diff_ext : forall x g, -SetRegReg.In x (interf_reg_reg g) \/ -SetRegReg.In x (pref_reg_reg g) -> fst x <> snd x. - -Proof. -Admitted. - -Lemma IE_reg_reg_diff : forall x y w, -EdgeSet.In (x,y,w) IE_reg_reg -> ~Vertex.eq x y. - -Proof. -intros. -unfold IE_reg_reg in H. -set (f := (fun (e : SetRegReg.elt) (s : EdgeSet.t) => - EdgeSet.add - (Regs.reg_to_Reg (fst e), Regs.reg_to_Reg (snd e), None) s)) in *. -rewrite SetRegReg.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize SetRegReg.elements_2. intro HH. -generalize (HH (interf_reg_reg g)). clear HH. intro HH. -induction (SetRegReg.elements (interf_reg_reg g)). simpl in H. -elim (EdgeSet.empty_1 H). -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f', f in H. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H). -fold (eq (Regs.reg_to_Reg (fst a), Regs.reg_to_Reg (snd a), None) (x,y,w)) in H0. -destruct (eq_charac _ _ H0); change_rewrite; destruct H1. -assert (fst a <> snd a). -apply set_reg_reg_diff_ext with (g:=g). -left. apply HH. -left. auto. -intro. rewrite <-H1 in H4. rewrite <-H2 in H4. -inversion H4. subst. elim (H3 H7). -assert (fst a <> snd a). -apply set_reg_reg_diff_ext with (g:=g). -left. apply HH. -left. auto. -intro. rewrite <-H1 in H4. rewrite <-H2 in H4. -inversion H4. subst. elim H3. auto. -apply IHl. auto. - -intros. apply HH. right. auto. - -unfold f', f. intros. -apply RegRegProps.add_add. - -unfold f', f. intros. -apply RegRegProps.Dec.F.add_m. apply E.eq_refl. auto. -Qed. - -Lemma IE_reg_mreg_diff : forall x y w, -EdgeSet.In (x,y,w) IE_reg_mreg -> ~Vertex.eq x y. - -Proof. -intros. -unfold IE_reg_mreg in H. -set (f := (fun (e : SetRegMreg.elt) (s : EdgeSet.t) => - EdgeSet.add - (Regs.reg_to_Reg (fst e), Regs.mreg_to_Reg (snd e), None) s)) in *. -rewrite SetRegMreg.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize SetRegMreg.elements_2. intro HH. -generalize (HH (interf_reg_mreg g)). clear HH. intro HH. -induction (SetRegMreg.elements (interf_reg_mreg g)). simpl in H. -elim (EdgeSet.empty_1 H). -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f', f in H. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H). -fold (eq (Regs.reg_to_Reg (fst a), Regs.mreg_to_Reg (snd a), None) (x,y,w)) in H0. -destruct (eq_charac _ _ H0); change_rewrite; destruct H1. -intro. rewrite <-H1 in H3. rewrite <-H2 in H3. inversion H3. -intro. rewrite <-H1 in H3. rewrite <-H2 in H3. inversion H3. -apply IHl. auto. - -intros. apply HH. right. auto. - -unfold f', f. intros. -apply RegRegProps.add_add. - -unfold f', f. intros. -apply RegRegProps.Dec.F.add_m. apply E.eq_refl. auto. -Qed. - -Lemma AE_reg_reg_diff : forall x y w, -EdgeSet.In (x,y,w) AE_reg_reg -> ~Vertex.eq x y. - -Proof. -intros. -unfold AE_reg_reg in H. -set (f := (fun (e : SetRegReg.elt) (s : EdgeSet.t) => - EdgeSet.add - (Regs.reg_to_Reg (fst e), Regs.reg_to_Reg (snd e), Some N0) s)) in *. -rewrite SetRegReg.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize SetRegReg.elements_2. intro HH. -generalize (HH (pref_reg_reg g)). clear HH. intro HH. -induction (SetRegReg.elements (pref_reg_reg g)). simpl in H. -elim (EdgeSet.empty_1 H). -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f', f in H. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H). -fold (eq (Regs.reg_to_Reg (fst a), Regs.reg_to_Reg (snd a), None) (x,y,w)) in H0. -destruct (eq_charac _ _ H0); change_rewrite; destruct H1. -assert (fst a <> snd a). -apply set_reg_reg_diff_ext with (g:=g). -right. apply HH. -left. auto. -intro. rewrite <-H1 in H4. rewrite <-H2 in H4. -inversion H4. subst. elim (H3 H7). -assert (fst a <> snd a). -apply set_reg_reg_diff_ext with (g:=g). -right. apply HH. -left. auto. -intro. rewrite <-H1 in H4. rewrite <-H2 in H4. -inversion H4. subst. elim H3. auto. -apply IHl. auto. - -intros. apply HH. right. auto. - -unfold f', f. intros. -apply RegRegProps.add_add. - -unfold f', f. intros. -apply RegRegProps.Dec.F.add_m. apply E.eq_refl. auto. -Qed. - -Lemma AE_reg_mreg_diff : forall x y w, -EdgeSet.In (x,y,w) AE_reg_mreg -> ~Vertex.eq x y. - -Proof. -intros. -unfold AE_reg_mreg in H. -set (f := (fun (e : SetRegMreg.elt) (s : EdgeSet.t) => - EdgeSet.add - (Regs.reg_to_Reg (fst e), Regs.mreg_to_Reg (snd e), Some N0) s)) in *. -rewrite SetRegMreg.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize SetRegMreg.elements_2. intro HH. -generalize (HH (pref_reg_mreg g)). clear HH. intro HH. -induction (SetRegMreg.elements (pref_reg_mreg g)). simpl in H. -elim (EdgeSet.empty_1 H). -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f', f in H. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H). -fold (eq (Regs.reg_to_Reg (fst a), Regs.mreg_to_Reg (snd a), None) (x,y,w)) in H0. -destruct (eq_charac _ _ H0); change_rewrite; destruct H1. -intro. rewrite <-H1 in H3. rewrite <-H2 in H3. inversion H3. -intro. rewrite <-H1 in H3. rewrite <-H2 in H3. inversion H3. -apply IHl. auto. - -intros. apply HH. right. auto. - -unfold f', f. intros. -apply RegRegProps.add_add. - -unfold f', f. intros. -apply RegRegProps.Dec.F.add_m. apply E.eq_refl. auto. -Qed. - -Lemma not_eq_extremities : forall x y, -VertexSet.In x (adj_set y interf_map) \/ VertexSet.In x (adj_set y pref_map) -> -~Vertex.eq x y. - -Proof. -intros. -destruct H. -generalize (set2map_charac _ _ _ _ H). intro. -destruct H0. destruct H0. -unfold interfgraph_interference_edges in H0. -destruct (EdgeSet.union_1 H0). -apply (IE_reg_reg_diff x y x0 H1). -apply (IE_reg_mreg_diff x y x0 H1). -rewrite adj_set_empty in H0. elim (VertexSet.empty_1 H0). -generalize (set2map_charac _ _ _ _ H). intro. -destruct H0. destruct H0. -unfold interfgraph_affinity_edges in H0. -generalize (resolve_conflicts_2 _ _ _ H0). intro. -clear H0. destruct H1 as [H0 _]. -destruct (EdgeSet.union_1 H0). -apply (AE_reg_reg_diff x y x0 H1). -apply (AE_reg_mreg_diff x y x0 H1). -rewrite adj_set_empty in H0. elim (VertexSet.empty_1 H0). -Qed. - -Definition graph_translation : new_graph := -let vert := interfgraph_vertices in -let iedges := interfgraph_interference_edges in -let aedges := resolve_conflicts (EdgeSet.union AE_reg_reg AE_reg_mreg) iedges in -let emp := VertexSet.fold - (fun x m => VertexMap.add x VertexSet.empty m) - vert - (VertexMap.empty VertexSet.t) in -let im := set2map iedges emp in -let pm := set2map aedges emp in -Make_Graph - vert im pm - extremities_imap - extremities_pmap - simple_graph - sym_imap - sym_pmap - not_eq_extremities. - -Lemma set2map_charac_2 : forall x y s m, -(exists w, EdgeSet.In (x,y,w) s) \/ VertexSet.In x (adj_set y m) -> -VertexSet.In x (adj_set y (set2map s m)). - -Proof. -intros. -unfold set2map. -set (f := (fun (e : EdgeSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add (fst_ext e) - (new_adj_set (fst_ext e) (snd_ext e) m) - (VertexMap.add (snd_ext e) - (new_adj_set (snd_ext e) (fst_ext e) m) m))) in *. -rewrite EdgeSet.fold_1. -set (f' := fun a e => f e a) in *. -destruct H. destruct H. -generalize EdgeSet.elements_1. intro HH. -generalize (HH s (x,y,x0) H). clear HH. intro HH. -induction (EdgeSet.elements s). -inversion HH. -cut (EqualSetMap (fold_left f' (a :: l) m) - (f' (fold_left f' l m) a)). intro. -generalize (H0 y). clear H0. intro H0. inversion H0. subst. -set (tmp := fold_left f' l m) in *. -unfold f', f in H3. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H3. -congruence. -apply Regs.eq_sym. auto. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_eq_o in H3. -congruence. -apply Regs.eq_sym. auto. -auto. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_neq_o in H3. -inversion HH; subst. -destruct (eq_charac _ _ H4); change_rewrite; destruct H4; destruct H1. -elim (n0 H6). -elim (n H6). -unfold adj_set in IHl. rewrite <-H3 in IHl. -elim (VertexSet.empty_1 (IHl H4)). -auto. -auto. - -set (tmp := fold_left f' l m) in *. -unfold adj_set. simpl. rewrite <-H1. rewrite H3. -unfold f',f in H2. -inversion HH; subst. -destruct (eq_charac _ _ H5); change_rewrite; destruct H4. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set. -destruct (VertexMap.find (fst_ext a) tmp). -apply VertexSet.add_1. rewrite H4. rewrite <-H6. rewrite <-e. apply Regs.eq_refl. -apply VertexSet.singleton_2. rewrite H4. rewrite <-H6. rewrite <-e. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set. destruct (VertexMap.find (snd_ext a) tmp). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -auto. -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set. destruct (VertexMap.find (fst_ext a) tmp). -apply VertexSet.add_1. apply Regs.eq_sym. auto. -apply VertexSet.singleton_2. apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set. -rewrite <-(MapFacts.find_o _ e). -case_eq (VertexMap.find y tmp); intros. -apply VertexSet.add_2. -unfold adj_set in IHl. rewrite H2 in IHl. -apply IHl. auto. -assert (VertexSet.In x VertexSet.empty). -unfold adj_set in IHl. rewrite H2 in IHl. apply IHl. auto. -elim (VertexSet.empty_1 H4). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set. -rewrite <-(MapFacts.find_o _ e). -case_eq (VertexMap.find y tmp); intros. -apply VertexSet.add_2. -unfold adj_set in IHl. rewrite H2 in IHl. apply IHl. auto. -assert (VertexSet.In x VertexSet.empty). -unfold adj_set in IHl. rewrite H2 in IHl. apply IHl. auto. -elim (VertexSet.empty_1 H4). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -unfold adj_set in IHl. rewrite <-H2 in IHl. apply IHl. auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold f', f, EqualSetMap. intros. -destruct (Vertex.eq_dec x1 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x1 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x1 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x1 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x1 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x1 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x1 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x1 s0). apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x1 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x1 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H0. auto. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x1 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x1 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x1 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold EqualSetMap, f', f. intros. -destruct (Vertex.eq_dec x1 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (fst_ext a0)). intro. -inversion H1. apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. -apply Regs.eq_refl. -auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x1 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (snd_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. -apply Regs.eq_refl. -auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -auto. -auto. -auto. -auto. - -induction (EdgeSet.elements s). simpl. auto. -cut (EqualSetMap (fold_left f' (a :: l) m) - (f' (fold_left f' l m) a)). intro. -generalize (H0 y). clear H0. intro H0. inversion H0. subst. -set (tmp := fold_left f' l m) in *. -unfold f', f in H3. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H3. -congruence. -apply Regs.eq_sym. auto. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_eq_o in H3. -congruence. -apply Regs.eq_sym. auto. -auto. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. rewrite <-H3 in IHl. elim (VertexSet.empty_1 IHl). -auto. -auto. - -unfold adj_set. simpl. rewrite <-H1. -rewrite H3. -set (tmp := fold_left f' l m) in *. -unfold f', f in H2. -destruct (Vertex.eq_dec y (fst_ext a)). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set. -rewrite <-(MapFacts.find_o _ e). -case_eq (VertexMap.find y tmp); intros. -apply VertexSet.add_2. -unfold adj_set in IHl. rewrite H2 in IHl. apply IHl. -unfold adj_set in IHl. rewrite H2 in IHl. inversion IHl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -destruct (Vertex.eq_dec y (snd_ext a)). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -unfold new_adj_set. -rewrite <-(MapFacts.find_o _ e). -case_eq (VertexMap.find y tmp); intros. -apply VertexSet.add_2. -unfold adj_set in IHl. rewrite H2 in IHl. apply IHl. -unfold adj_set in IHl. rewrite H2 in IHl. inversion IHl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -unfold adj_set in IHl. rewrite <-H2 in IHl. apply IHl. -auto. -auto. - -apply fold_left_assoc_map. -unfold f', f, EqualSetMap. intros. -destruct (Vertex.eq_dec x0 (fst_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext z)). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -rewrite <-(MapFacts.find_o _ e). -rewrite <-(MapFacts.find_o _ e0). -destruct (VertexMap.find x0 s0). -apply Props.add_add. -rewrite Props.singleton_equal_add. apply Props.add_add. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n. rewrite H0. auto. -rewrite <-e. rewrite <-e0. apply Regs.eq_refl. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n1. rewrite H0. auto. -intro. elim n0. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (fst_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext y0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply VertexSet.eq_refl. -intro. elim n0. rewrite H0. auto. -intro. elim n. rewrite H0. auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. -auto. -auto. -auto. -auto. - -intros. -unfold EqualSetMap, f', f. intros. -destruct (Vertex.eq_dec x0 (fst_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (fst_ext a0)). intro. -inversion H1. apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. -apply Regs.eq_refl. -auto. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 (snd_ext a0)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -unfold new_adj_set. -generalize (H0 (snd_ext a0)). intro. -inversion H1. -apply VertexSet.eq_refl. -apply Props.Dec.F.add_m. -apply Regs.eq_refl. -auto. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -auto. -auto. -auto. -auto. -auto. -Qed. - -Lemma regreg_IE : forall x y, -SetRegReg.In (x,y) (interf_reg_reg g) -> -EdgeSet.In (Regs.reg_to_Reg x,Regs.reg_to_Reg y,None) IE_reg_reg. - -Proof. -intros. -unfold IE_reg_reg. -set (f := (fun (e : SetRegReg.elt) (s : EdgeSet.t) => - EdgeSet.add (Regs.reg_to_Reg (fst e), Regs.reg_to_Reg (snd e), None) s)) in *. -rewrite SetRegReg.fold_1. -set (f' := fun a e => f e a) in *. -generalize SetRegReg.elements_1. intro HH. -generalize (HH (interf_reg_reg g) (x,y) H). clear HH. intro HH. -induction (SetRegReg.elements (interf_reg_reg g)). simpl. -inversion HH. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f', f. -inversion HH; subst. -apply EdgeSet.add_1. destruct H1. simpl in *. subst. apply Edge.eq_refl. -apply EdgeSet.add_2. apply IHl. auto. -unfold f', f. intros. -apply RegRegProps.add_add. - -unfold f', f. intros. -apply RegRegProps.Dec.F.add_m. apply E.eq_refl. auto. -Qed. - -Lemma regmreg_IE : forall x y, -SetRegMreg.In (x,y) (interf_reg_mreg g) -> -EdgeSet.In (Regs.reg_to_Reg x,Regs.mreg_to_Reg y,None) IE_reg_mreg. - -Proof. -intros. -unfold IE_reg_mreg. -set (f := (fun (e : SetRegMreg.elt) (s : EdgeSet.t) => - EdgeSet.add (Regs.reg_to_Reg (fst e), Regs.mreg_to_Reg (snd e), None) s)) in *. -rewrite SetRegMreg.fold_1. -set (f' := fun a e => f e a) in *. -generalize SetRegMreg.elements_1. intro HH. -generalize (HH (interf_reg_mreg g) (x,y) H). clear HH. intro HH. -induction (SetRegMreg.elements (interf_reg_mreg g)). simpl. -inversion HH. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f', f. -inversion HH; subst. -apply EdgeSet.add_1. destruct H1. simpl in *. subst. apply Edge.eq_refl. -apply EdgeSet.add_2. apply IHl. auto. -unfold f', f. intros. -apply RegRegProps.add_add. - -unfold f', f. intros. -apply RegRegProps.Dec.F.add_m. apply E.eq_refl. auto. -Qed. - -Lemma regreg_IE_translation : forall x y, -SetRegReg.In (x,y) (interf_reg_reg g) -> -EdgeSet.In (Regs.P x,Regs.P y,None) (IE graph_translation). - -Proof. -intros. -unfold graph_translation. unfold IE. simpl. -rewrite (edgemap_to_edgeset_charac _ _ _ _ sym_imap). -apply set2map_charac_2. -left. exists None. -unfold interfgraph_interference_edges. apply EdgeSet.union_2. -rewrite edge_comm. apply regreg_IE. auto. -Qed. - -Lemma regmreg_IE_translation : forall x y, -SetRegMreg.In (x,y) (interf_reg_mreg g) -> -EdgeSet.In (Regs.P x,Regs.M y,None) (IE graph_translation). - -Proof. -intros. -unfold graph_translation. unfold IE. simpl. -rewrite (edgemap_to_edgeset_charac _ _ _ _ sym_imap). -apply set2map_charac_2. -left. exists None. -unfold interfgraph_interference_edges. apply EdgeSet.union_3. -rewrite edge_comm. apply regmreg_IE. auto. -Qed. - -End Translation.
\ No newline at end of file diff --git a/backend/IRC.v b/backend/IRC.v deleted file mode 100755 index 7842a4b..0000000 --- a/backend/IRC.v +++ /dev/null @@ -1,122 +0,0 @@ -Require Import Recdef. -Require Import FSetInterface. -Require Import InterfGraphMapImp. -Require Import OrderedOption. -Require Import FMapAVL. -Require Import IRC_termination. -Require Import IRC_graph. -Require Import IRC_Graph_Functions. -Require Import Edges. - -Import Edge RegFacts Props OTFacts. - -(* Definition of Register options *) - -Module OptionReg := OrderedOpt Register. - -(* Definition of the type of colorings *) - -Definition Coloring := Register.t -> OptionReg.t. - -(* Map used to build a coloring of the graph. - A coloring is a the function find applied to the map *) - -Module ColorMap := FMapAVL.Make Register. - -(* Function to complete a coloring by giving the - same color to coalesced vertices *) - -Definition complete_coloring e col : ColorMap.t Register.t := -let x := snd_ext e in let y := fst_ext e in -match ColorMap.find y col with -| None => col -| Some c => ColorMap.add x c col -end. - -(* Function to compute forbidden colors when completing the coloring - for simplified or spilled registers (optimistic spilling) *) - -Section add_monad. - -Definition vertex_add_monad (o : option Register.t) (VS : VertexSet.t) := -match o with -|Some v => VertexSet.add v VS -|None => VS -end. - -Variable A : Type. - -Lemma monad_fold : forall f a l s, -VertexSet.Equal (fold_left - (fun (x : VertexSet.t) (e : A) => vertex_add_monad (f e) x) - (a :: l) s) - (vertex_add_monad (f a) - (fold_left (fun (x : VertexSet.t) (e : A) => vertex_add_monad (f e) x) l s)). - -Proof. -intros f a l s;apply fold_left_assoc. -intros y z e;unfold vertex_add_monad. -destruct (f y);destruct (f z);[apply add_add|intuition|intuition|intuition]. -intros e1 e2 y H. unfold vertex_add_monad. -destruct (f y);intuition. -apply Dec.F.add_m; intuition. -Qed. - -End add_monad. - -Definition forbidden_colors x col g := -VertexSet.fold (fun v => vertex_add_monad (ColorMap.find v col)) - (interference_adj x g) - VertexSet.empty. - -(* Function to complete the coloring for simplified or spilled vertices. - Calls forbidden_colors. Is choosing yet any available color *) - -Definition available_coloring ircg x (col : ColorMap.t Register.t) := -let g := (irc_g ircg) in let palette := (pal ircg) in -match (VertexSet.choose (VertexSet.diff palette (forbidden_colors x col g))) with -| None => col -| Some c => ColorMap.add x c col -end. - -(* Definition of the empty coloring as the coloring where the only - colored vertices are the precolored ones *) - -Definition precoloring_map g := -VertexSet.fold (fun x => ColorMap.add x x) (precolored g) (ColorMap.empty Register.t). - -Function IRC_map (g : irc_graph) -{measure irc_measure g} : ColorMap.t Register.t := -match simplify g with -|Some rg' => let (r,g') := rg' in available_coloring g r (IRC_map g') -|None => match coalesce g with - |Some eg' => let (e,g') := eg' in complete_coloring e (IRC_map g') - |None => match freeze g with - |Some rg' => let (r,g') := rg' in IRC_map g' - |None => match spill g with - |Some rg' => let (r,g') := rg' in available_coloring g r (IRC_map g') - |None => precoloring_map (irc_g g) - end - end - end -end. - -Proof. -intros. apply (simplify_dec g r g' teq). -intros. apply (coalesce_dec g e g' teq0). -intros. apply (freeze_dec g r g' teq1). -intros. apply (spill_dec g r g' teq2). -Qed. - -(* Definition of the transformation from a map to a coloring, - simply by using find *) - -Definition map_to_coloring (colmap : ColorMap.t Register.t) := -fun x => ColorMap.find x colmap. - -(* Final definition of iterated register coalescing *) - -Definition IRC g palette := -let g' := graph_to_IRC_graph g palette in map_to_coloring (IRC_map g'). - -Definition precoloring g := map_to_coloring (precoloring_map g). diff --git a/backend/IRCColoring.v b/backend/IRCColoring.v deleted file mode 100755 index 5efeed7..0000000 --- a/backend/IRCColoring.v +++ /dev/null @@ -1,847 +0,0 @@ -Require Import Recdef. -Require Import IRC. -Require Import FSetInterface. -Require Import Edges. -Require Import Interference_adjacency. -Require Import IRC_graph. -Require Import Conservative_criteria. -Require Import InterfGraphMapImp. -Require Import Graph_Facts. -Require Import IRC_Graph_Functions. -Require Import WS. - -Import Edge RegFacts Props OTFacts. - -Module MapFacts := FMapFacts.Facts ColorMap. - -Definition proper_coloring_1 (col : Coloring) g := forall e x y, -interf_edge e -> -In_graph_edge e g -> -OptionReg.eq (col (fst_ext e)) (Some x) -> -OptionReg.eq (col (snd_ext e)) (Some y) -> -~Register.eq x y. - -Definition proper_coloring_2 (col : Coloring) g := forall x y, -OptionReg.eq (col x) (Some y) -> In_graph x g. - -Definition proper_coloring_3 (col : Coloring) palette := forall x y, -OptionReg.eq (col x) (Some y) -> VertexSet.In y palette. - -Definition proper_coloring col g palette := -proper_coloring_1 col g /\ proper_coloring_2 col g /\ proper_coloring_3 col palette. - -Lemma diff_empty_sub : forall s1 s2, -VertexSet.Empty (VertexSet.diff s1 s2) -> -VertexSet.Subset s1 s2. - -Proof. -intros s1 s2 H. -intros a H0. -destruct (In_dec a s2). -assumption. -elim (H a). -apply (VertexSet.diff_3 H0 n). -Qed. - -Lemma interf_in_forbidden : forall x y col g c, -Interfere x y g -> -OptionReg.eq (map_to_coloring col y) (Some c) -> -VertexSet.In c (forbidden_colors x col g). - -Proof. -intros x y col g c H H0. -unfold forbidden_colors. -unfold Interfere in H. rewrite <-in_interf in H. -generalize (interf_adj_comm _ _ _ H). intro H1. -generalize (VertexSet.elements_1 H1);clear H1;intro H1. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (interference_adj x g)). -inversion H1. -rewrite monad_fold. -inversion H1;subst. -unfold map_to_coloring in H0. -inversion H0;subst. -rewrite (MapFacts.find_o col H3) in H2. -rewrite <-H2. -simpl;apply VertexSet.add_1. -assumption. -destruct (ColorMap.find a col);[apply VertexSet.add_2|];apply IHl;assumption. -Qed. - -Lemma available_coloring_1 : forall col x g, -proper_coloring (map_to_coloring col) (irc_g g) (pal g) -> -In_graph x (irc_g g) -> -proper_coloring (map_to_coloring (available_coloring g x col)) (irc_g g) (pal g). - -Proof. -intros col x ircg H HH. unfold available_coloring. -set (palette := pal ircg) in *. set (g := irc_g ircg) in *. -case_eq (VertexSet.choose (VertexSet.diff palette (forbidden_colors x col g))). -intros c H1. -generalize H1;intro H0. -unfold proper_coloring. -split. -unfold proper_coloring_1. -intros e x' y' H2 H3 H4 H5. -destruct (Register.eq_dec x (fst_ext e)). -unfold map_to_coloring in H4;unfold map_to_coloring in H5. -rewrite MapFacts.add_eq_o in H4. -rewrite MapFacts.add_neq_o in H5. -generalize (VertexSet.choose_1 H1);clear H1;intro H1. -generalize (VertexSet.diff_1 H1);intro H6. -generalize (VertexSet.diff_2 H1);intro H7;clear H1. -assert (Interfere x (snd_ext e) g) as Hinterf. -unfold Interfere. -assert (eq (fst_ext e, snd_ext e,None) (x, snd_ext e, None)). -Eq_eq. -rewrite (edge_eq e) in H3. -rewrite H2 in H3. -rewrite H1 in H3;assumption. -generalize (interf_in_forbidden x (snd_ext e) col g y' Hinterf H5);intro H1. -intro H8. -inversion H4;subst. -generalize (Register.eq_trans H11 H8);clear H11 H8;intro H8. -rewrite H8 in H7. -elim (H7 H1). -intro Heq. -elim (In_graph_edge_diff_ext _ _ H3). -apply (Register.eq_trans (Register.eq_sym Heq) e0). -assumption. -destruct (Register.eq_dec x (snd_ext e)). -unfold map_to_coloring in H4;unfold map_to_coloring in H5. -rewrite MapFacts.add_eq_o in H5. -rewrite MapFacts.add_neq_o in H4. -generalize (VertexSet.choose_1 H1);clear H1;intro H1. -generalize (VertexSet.diff_1 H1);intro H6. -generalize (VertexSet.diff_2 H1);intro H7;clear H1. -assert (Interfere x (fst_ext e) g) as Hinterf. -unfold Interfere. -rewrite (edge_eq e) in H3. rewrite H2 in H3. -assert (eq (fst_ext e, snd_ext e,None) (x, fst_ext e, None)) by Eq_comm_eq. -rewrite H1 in H3;assumption. -generalize (interf_in_forbidden x (fst_ext e) col g x' Hinterf H4);intro H1. -intro H8. -inversion H5;subst. -generalize (Register.eq_trans H11 (Register.eq_sym H8));clear H11 H8;intro H8. -rewrite H8 in H7. -elim (H7 H1). -intro Heq. -elim (In_graph_edge_diff_ext _ _ H3). -apply (Register.eq_trans (Register.eq_sym e0) Heq). -assumption. -unfold map_to_coloring in H4;unfold map_to_coloring in H5. -rewrite MapFacts.add_neq_o in H5. -rewrite MapFacts.add_neq_o in H4. -unfold proper_coloring in H;destruct H as [H _]. -unfold proper_coloring_1 in H. -apply (H e);assumption. -assumption. -assumption. -split. -unfold proper_coloring in *. -destruct H as [_ H];destruct H as [H _]. -unfold proper_coloring_2 in *. -intros x' y' H2. -destruct (Register.eq_dec x x'). -unfold map_to_coloring in H2. -rewrite e in HH;assumption. -apply (H x' y'). -unfold map_to_coloring in H2. -rewrite MapFacts.add_neq_o in H2. -assumption. -assumption. -unfold proper_coloring in *. -do 2 destruct H as [_ H]. -unfold proper_coloring_3 in *. -intros x' y' H2. -destruct (Register.eq_dec x x'). -unfold map_to_coloring in H2. -generalize (VertexSet.choose_1 H1);clear H1;intro H1. -generalize (VertexSet.diff_1 H1);intro H3. -rewrite MapFacts.add_eq_o in H2. -inversion H2;subst. -rewrite H6 in H3;assumption. -assumption. -apply (H x' y'). -unfold map_to_coloring in H2. -rewrite MapFacts.add_neq_o in H2. -assumption. -assumption. -auto. -Qed. - -Lemma complete_coloring_1 : forall col e g, -In_graph_edge e g -> -ColorMap.find (snd_ext e) col = None -> -OptionReg.eq -(map_to_coloring (complete_coloring e col) (snd_ext e)) -(map_to_coloring (complete_coloring e col) (fst_ext e)). - -Proof. -intros col e g HH Hin;unfold complete_coloring. -case_eq (ColorMap.find (fst_ext e) col). -intros r H. -unfold map_to_coloring. -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite H;apply OptionReg.eq_refl. -apply (In_graph_edge_diff_ext _ _ HH). -apply Register.eq_refl. -intro H0. -unfold map_to_coloring. -rewrite Hin;rewrite H0;apply OptionReg.eq_refl. -Qed. - -Lemma complete_coloring_2 : forall col e x, -~Register.eq (snd_ext e) x -> -OptionReg.eq -(map_to_coloring (complete_coloring e col) x) -(map_to_coloring col x). - -Proof. -intros col e x H. -unfold complete_coloring. -destruct (ColorMap.find (fst_ext e) col). -unfold map_to_coloring. -rewrite MapFacts.add_neq_o. -apply OptionReg.eq_refl. -assumption. -apply OptionReg.eq_refl. -Qed. - -Definition compat_col col := forall x y, -Register.eq x y -> OptionReg.eq (col x) (col y). - -Lemma compat_IRC : forall g palette, compat_col (IRC g palette). - -Proof. -unfold IRC. -unfold compat_col. -intros g palette x y H. -unfold map_to_coloring. -rewrite MapFacts.find_o with (y := y). -apply OptionReg.eq_refl. -assumption. -Qed. - -Lemma compat_complete : forall col e, -compat_col (map_to_coloring col) -> -compat_col (map_to_coloring (complete_coloring e col)). - -Proof. -unfold compat_col;intros col e H. -intros x y Heq;generalize (H x y Heq);clear H;intro H. -unfold complete_coloring. -destruct (ColorMap.find (fst_ext e) col). -unfold map_to_coloring. -destruct (Register.eq_dec x (snd_ext e)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -apply OptionReg.eq_refl. -apply (Register.eq_trans (Register.eq_sym e0) Heq). intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -assumption. -intro Helim;elim n. -apply (Register.eq_trans Heq (Register.eq_sym Helim)). -intro H0;elim n;auto. -assumption. -Qed. - -Lemma colored_complete_diff_colored : forall e col, -~Register.eq (snd_ext e) (fst_ext e) -> -OptionReg.eq (map_to_coloring (complete_coloring e col) (fst_ext e)) (map_to_coloring col (fst_ext e)). - -Proof. -intros e col H. -unfold complete_coloring. -case_eq (ColorMap.find (fst_ext e) col). -intros r H0. -unfold map_to_coloring. -rewrite MapFacts.add_neq_o. -apply OptionReg.eq_refl. -assumption. -intro H0. -apply OptionReg.eq_refl. -Qed. - -Lemma complete_coloring_coloring : forall col e g Hin Haff, -~VertexSet.In (snd_ext e) (precolored (irc_g g)) -> -proper_coloring (map_to_coloring col) (merge e (irc_g g) Hin Haff) (pal g) -> -conservative_coalescing_criteria e (irc_g g) (irc_k g) = true -> -compat_col (map_to_coloring col) -> -proper_coloring (map_to_coloring (complete_coloring e col)) (irc_g g) (pal g). - -Proof. -unfold proper_coloring;unfold proper_coloring_1; -unfold proper_coloring_2;unfold proper_coloring_3. -intros col e ircg H1 Haff Hpre H H0 Hcompat. -set (g := irc_g ircg) in *. set (palette := pal ircg) in *. -rewrite <-(Hk ircg) in *. -destruct H as [H HH];destruct HH as [HH HHH]. -split. -intros e' x' y' H2 H3 H4 H5. -apply (H (redirect (snd_ext e) (fst_ext e) e')). -unfold interf_edge;rewrite redirect_weight_eq;assumption. -apply In_merge_interf_edge. -assumption. -assumption. -(*intro Heq. -generalize (get_weight_m _ _ Heq);intro H6. -rewrite H2 in H6. -unfold aff_edge in Haff. -destruct Haff as [w Haff]. -rewrite Haff in H6. -inversion H6. -assumption.*) -destruct e' as [e' we'];destruct e' as [e'1 e'2]. -unfold redirect. -unfold interf_edge in H2;simpl in H2;subst. -change_rewrite. -destruct (OTFacts.eq_dec e'1 (snd_ext e));change_rewrite. -generalize (compat_complete col e);intro Hcompat2. -generalize (Hcompat2 Hcompat _ _ r);intro H7. -generalize (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H4) H7); -clear H7;intro H7. -assert (ColorMap.find (snd_ext e) col = None) as Hn. -case_eq (ColorMap.find (snd_ext e) col). -clear H3 H4 H5 r Hcompat2 H7. -intros r H8. -generalize (HH (snd_ext e) r);intro Helim. -unfold map_to_coloring in Helim. -rewrite H8 in Helim. -generalize (Helim (OptionReg.eq_refl _));clear Helim;intro Helim. -rewrite In_merge_vertex in Helim. destruct Helim. elim H3. auto. -auto. -generalize (complete_coloring_1 col e g H1 Hn);intro H8. -generalize (OptionReg.eq_trans _ _ _ H7 H8);intro H9. -generalize (complete_coloring_2 col e (fst_ext e) (In_graph_edge_diff_ext _ _ H1)). -intro H10. -generalize (OptionReg.eq_trans _ _ _ H9 H10);intro H11. -apply OptionReg.eq_sym;assumption. -destruct (OTFacts.eq_dec e'2 (snd_ext e));change_rewrite; -apply OptionReg.eq_trans with (y := (map_to_coloring (complete_coloring e col) e'1)); -[apply OptionReg.eq_sym;apply complete_coloring_2;intuition|assumption| -apply OptionReg.eq_sym;apply complete_coloring_2;intuition|assumption]. -destruct e' as [e' we'];destruct e' as [e'1 e'2]. -unfold redirect;change_rewrite. -unfold interf_edge in H2;simpl in H2;subst. -destruct (OTFacts.eq_dec e'1 (snd_ext e));change_rewrite. -destruct (Register.eq_dec (snd_ext e) e'2). -elim (In_graph_edge_diff_ext _ _ H3). -apply (Register.eq_trans (Register.eq_sym e0) (Register.eq_sym r)). -generalize (complete_coloring_2 col e e'2 n);intro H6. -apply OptionReg.eq_trans with (y := map_to_coloring (complete_coloring e col) e'2). -apply OptionReg.eq_sym;assumption. -assumption. -destruct (OTFacts.eq_dec e'2 (snd_ext e));change_rewrite. -generalize (compat_complete col e);intro Hcompat2. -generalize (Hcompat2 Hcompat _ _ r);intro H7. -generalize (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H5) H7); -clear H7;intro H7. -assert (ColorMap.find (snd_ext e) col = None) as Hn. -case_eq (ColorMap.find (snd_ext e) col). -clear H3 H4 H5 r Hcompat2 H7. -intros r H8. -generalize (HH (snd_ext e) r);intro Helim. -unfold map_to_coloring in Helim. -rewrite H8 in Helim. -generalize (Helim (OptionReg.eq_refl _));clear Helim;intro Helim. -rewrite In_merge_vertex in Helim. destruct Helim. elim H3. auto. -auto. -generalize (complete_coloring_1 col e g H1 Hn);intro H8. -generalize (OptionReg.eq_trans _ _ _ H7 H8);intro H9. -generalize (complete_coloring_2 col e (fst_ext e) (In_graph_edge_diff_ext _ _ H1)). -intro H10. -generalize (OptionReg.eq_trans _ _ _ H9 H10);intro H11. -apply OptionReg.eq_sym;assumption. -change (snd_ext (e'1,e'2,None)) with e'2. -assert (~Register.eq (snd_ext e) e'2) by intuition. -generalize (complete_coloring_2 col e e'2 H2);intro H6. -apply OptionReg.eq_trans with (y := map_to_coloring (complete_coloring e col) e'2). -apply OptionReg.eq_sym;assumption. -assumption. -(* second step *) -split. -intros x y H2. -destruct (Register.eq_dec x (snd_ext e)). -rewrite e0. -apply (In_graph_edge_in_ext _ _ H1). -assert (In_graph x (merge e g H1 Haff)). -apply HH with (y := y). -apply OptionReg.eq_trans with (y := map_to_coloring (complete_coloring e col) x). -apply OptionReg.eq_sym. -apply complete_coloring_2. -intuition. -assumption. -rewrite In_merge_vertex in H3. intuition. -(* third step *) -intros x y H2. -destruct (Register.eq_dec x (snd_ext e)). -assert (ColorMap.find (snd_ext e) col = None) as Hn. -case_eq (ColorMap.find (snd_ext e) col). -intros r H8. -generalize (HH (snd_ext e) r);intro Helim. -unfold map_to_coloring in Helim. -rewrite H8 in Helim. -generalize (Helim (OptionReg.eq_refl _));clear Helim;intro Helim. -rewrite In_merge_vertex in Helim. destruct Helim. elim H4. auto. -auto. -generalize (complete_coloring_1 col e g H1 Hn);intro H3. -generalize (complete_coloring_2 col e (fst_ext e) (In_graph_edge_diff_ext _ _ H1)). -intro H4. -generalize (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H4) (OptionReg.eq_sym _ _ H3)). -clear H3 H4;intro H3. -generalize (compat_complete col e Hcompat);intro Hcompat2. -generalize (Hcompat2 _ _ e0). -intro H4. -generalize (OptionReg.eq_trans _ _ _ H3 (OptionReg.eq_sym _ _ H4)). -clear H3 H4;intro H3. -apply HHH with (x:= fst_ext e). -apply (OptionReg.eq_trans _ _ _ H3 H2). -assert (~Register.eq (snd_ext e) x) by intuition. -generalize (complete_coloring_2 col e x H3). -intro H4. -apply HHH with (x:= x). -apply (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H4) H2). -Qed. - -Lemma interf_edge_in_delete_preference : forall x e g H, -interf_edge e -> -In_graph_edge e g -> -In_graph_edge e (delete_preference_edges x g H). - -Proof. -intros x e g HH H H0. -generalize (proj2 (In_graph_interf_edge_in_IE _ _) (conj H H0));intro H1. -rewrite In_delete_preference_edges_edge. -split. assumption. -intro. destruct H2. destruct H2. congruence. -Qed. - -Lemma coloring_delete_preference : forall col x g H palette, -proper_coloring col (delete_preference_edges x g H) palette -> -proper_coloring col g palette. - -Proof. -unfold proper_coloring;unfold proper_coloring_1; -unfold proper_coloring_2;unfold proper_coloring_3. -intros col x g Hdp palette H. -destruct H as [H HH];destruct HH as [HH HHH]. -split. -intros e x' y' H0 H1 H2 H3. -apply H with (e:=e). -assumption. -apply interf_edge_in_delete_preference;assumption. -assumption. -assumption. -split. -intros x' y' H0. rewrite <-(In_delete_preference_edges_vertex x' x g Hdp). -apply (HH x' y');auto. -assumption. -Qed. - -Lemma proper_remove_proper : forall col g x palette, -proper_coloring col (remove_vertex x g) palette -> -compat_col col -> -proper_coloring col g palette. - -Proof. -intros col g x palette H Hcompat. -unfold proper_coloring in *;unfold proper_coloring_1 in *; -unfold proper_coloring_2 in *;unfold proper_coloring_3 in *. -destruct H as [H HH];destruct HH as [HH HHH]. -split. -intros e x' y' H0 H1 H2 H3. -destruct (incident_dec e x). -destruct H4. -generalize (Hcompat _ _ H4);intro H5. -generalize (OptionReg.eq_trans _ _ _ H5 H2);clear H5;intro H5. -generalize (HH _ _ H5). intro. -rewrite In_remove_vertex in H6. destruct H6. elim H7. auto. -generalize (Hcompat _ _ H4);intro H5. -generalize (OptionReg.eq_trans _ _ _ H5 H3);clear H5;intro H5. -generalize (HH _ _ H5). intro. -rewrite In_remove_vertex in H6. destruct H6. elim H7. auto. -apply H with (e:=e); auto. -rewrite In_remove_edge; auto. -split. -intros x' y' H0. -generalize (HH x' y' H0);intro H1. -rewrite In_remove_vertex in H1. intuition. -assumption. -Qed. - -Section Fold_Facts. - -Variable A : Type. - -Lemma fold_left_compat_map : forall (f : ColorMap.t Register.t -> A -> ColorMap.t Register.t) l e e', -ColorMap.Equal e e' -> -(forall e1 e2 a, ColorMap.Equal e1 e2 -> ColorMap.Equal (f e1 a) (f e2 a)) -> -ColorMap.Equal (fold_left f l e) (fold_left f l e'). - -Proof. -intros f l. -induction l;simpl. -auto. -intros e e' H H0 H1. -apply (IHl (f e a) (f e' a)). -apply H0;assumption. -assumption. -Qed. - -End Fold_Facts. - -Lemma empty_coloring_simpl : forall l a, -NoDupA Register.eq (a :: l) -> -ColorMap.Equal (fold_left (fun (a0 : ColorMap.t VertexSet.elt) (e : VertexSet.elt) => - ColorMap.add e e a0) (a :: l) (ColorMap.empty Register.t)) - (ColorMap.add a a ( - fold_left (fun (a0 : ColorMap.t VertexSet.elt) (e : VertexSet.elt) => - ColorMap.add e e a0) l (ColorMap.empty Register.t))). - -Proof. -intro l;generalize (ColorMap.empty Register.t). -induction l;simpl;intros. -unfold ColorMap.Equal;auto. -assert (ColorMap.Equal (ColorMap.add a a (ColorMap.add a0 a0 t0)) - (ColorMap.add a0 a0 (ColorMap.add a a t0))). -unfold ColorMap.Equal. -intro y. -destruct (Register.eq_dec a0 a). -inversion H;subst. -elim H2. -left;auto. -destruct (Register.eq_dec y a). -destruct (Register.eq_dec y a0). -elim (n (Register.eq_trans (Register.eq_sym e0) e)). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -reflexivity. -intuition. -intro Hneq;elim n0;auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Register.eq_dec a0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -reflexivity. -assumption. -assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -reflexivity. -intro Hneq;elim n0;auto. -assumption. -assumption. -intro Hneq;elim n0;auto. -rewrite (fold_left_compat_map _ - (fun (a1 : ColorMap.t VertexSet.elt) (e : VertexSet.elt) => ColorMap.add e e a1) l _ _ H0). -simpl in IHl;apply IHl. -constructor. -inversion H;subst. -intro H5;elim H3. -right;auto. -inversion H;subst. -inversion H4;subst;assumption. -intros e1 e2 a1 H1. -unfold ColorMap.Equal;intro y. -destruct (Register.eq_dec a1 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -reflexivity. -assumption. -assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. -assumption. -assumption. -Qed. - -Lemma map_to_coloring_ext : forall c1 c2 g palette, -ColorMap.Equal c2 c1 -> -proper_coloring (map_to_coloring c1) g palette -> -proper_coloring (map_to_coloring c2) g palette. - -Proof. -unfold proper_coloring;unfold proper_coloring_1; -unfold proper_coloring_2;unfold proper_coloring_3. -intros c1 c2 g palette H H0. -destruct H0 as [H0 H1];destruct H1 as [H1 H2]. -split. -intros e x y H3 H4 H5 H6. -apply (H0 e);try assumption. -unfold ColorMap.Equal in H. -apply OptionReg.eq_trans with (y := (map_to_coloring c2 (fst_ext e))). -unfold map_to_coloring. -rewrite H. -apply OptionReg.eq_refl. -assumption. -apply OptionReg.eq_trans with (y := (map_to_coloring c2 (snd_ext e))). -unfold map_to_coloring. -rewrite H. -apply OptionReg.eq_refl. -assumption. -split. -intros x y H3. -apply (H1 x y). -apply OptionReg.eq_trans with (y := (map_to_coloring c2 x)). -unfold map_to_coloring. -rewrite H. -apply OptionReg.eq_refl. -assumption. -intros x y H3. -apply (H2 x y). -apply OptionReg.eq_trans with (y := (map_to_coloring c2 x)). -unfold map_to_coloring. -rewrite H. -apply OptionReg.eq_refl. -assumption. -Qed. - -Lemma compat_map_to_coloring : forall c1 c2 x, -ColorMap.Equal c1 c2 -> -OptionReg.eq (map_to_coloring c1 x) (map_to_coloring c2 x). - -Proof. -intros c1 c2 x H;unfold map_to_coloring; -unfold ColorMap.Equal in H. -rewrite H;apply OptionReg.eq_refl. -Qed. - -Lemma proper_coloring_empty_aux : forall g x, -OptionReg.eq (precoloring g x) (Some x) \/ -OptionReg.eq (precoloring g x) None. - -Proof. -intros g x. -unfold precoloring, precoloring_map. -rewrite VertexSet.fold_1. - -assert (NoDupA Register.eq (VertexSet.elements (precolored g))) as HNoDup by - (apply RegFacts.NoDupA_elements). -induction (VertexSet.elements (precolored g)). -simpl. -right;unfold map_to_coloring;rewrite MapFacts.empty_o. -apply OptionReg.eq_refl. -(* induction case *) -generalize (empty_coloring_simpl l a HNoDup);intro H0. -inversion HNoDup;subst. -generalize (IHl H3);clear IHl H2 H3;intro IHl. -destruct IHl. -left. -generalize (compat_map_to_coloring _ _ x H0);clear H0;intro H0. -apply (OptionReg.eq_trans _ _ _ H0). -destruct (Register.eq_dec a x). -unfold map_to_coloring. -rewrite MapFacts.add_eq_o. -constructor;assumption. -assumption. -unfold map_to_coloring. -rewrite MapFacts.add_neq_o. -apply H. -assumption. -destruct (Register.eq_dec a x). -left. -generalize (compat_map_to_coloring _ _ x H0);clear H0;intro H0. -apply (OptionReg.eq_trans _ _ _ H0). -unfold map_to_coloring. -rewrite MapFacts.add_eq_o. -constructor;assumption. -assumption. -right. -generalize (compat_map_to_coloring _ _ x H0);clear H0;intro H0. -apply (OptionReg.eq_trans _ _ _ H0). -unfold map_to_coloring. -rewrite MapFacts.add_neq_o. -apply H. -assumption. -Qed. - -Lemma in_empty_coloring_in_precolored : forall x g y, -OptionReg.eq (precoloring g x)(Some y) -> -VertexSet.In x (precolored g). - -Proof. -intros x g y H. unfold precoloring, precoloring_map in H. -rewrite VertexSet.fold_1 in H. -generalize (VertexSet.elements_2);intro H0. -generalize (H0 (precolored g));clear H0;intro H0. -assert (NoDupA Register.eq (VertexSet.elements (precolored g))) as HNoDup by - (apply RegFacts.NoDupA_elements). -induction (VertexSet.elements (precolored g)). -simpl in H. -unfold map_to_coloring in H;rewrite MapFacts.empty_o in H. -inversion H. -destruct (Register.eq_dec a x). -apply H0. -left; intuition. -apply IHl. -generalize (empty_coloring_simpl l a HNoDup);intro H1. -generalize (compat_map_to_coloring _ _ x H1);clear H1;intro H1. -unfold map_to_coloring in H1. -rewrite MapFacts.add_neq_o in H1. -apply (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H1)). -clear H1 IHl. -assumption. -assumption. -intros;apply H0;right;auto. -inversion HNoDup;auto. -Qed. - -Lemma proper_coloring_pre_inv : forall g x, -VertexSet.In x (precolored g) -> -OptionReg.eq (precoloring g x) (Some x). - -Proof. -intros g x H. unfold precoloring, precoloring_map. -rewrite VertexSet.fold_1. -generalize (VertexSet.elements_1);intro H0. -generalize (H0 (precolored g) x);clear H0;intro H0. -assert (NoDupA Register.eq (VertexSet.elements (precolored g))) as HNoDup by - (apply RegFacts.NoDupA_elements). -induction (VertexSet.elements (precolored g)). -generalize (H0 H). intro. inversion H1. -generalize (empty_coloring_simpl l a HNoDup);intro H1. simpl in *. -generalize (compat_map_to_coloring _ _ x H1). intro. -unfold map_to_coloring in H2. -destruct (Register.eq_dec x a). - rewrite MapFacts.add_eq_o in H2. -apply (OptionReg.eq_trans _ _ _ H2). -constructor. apply Register.eq_sym. auto. -apply Register.eq_sym. auto. -rewrite (MapFacts.add_neq_o) in H2. apply (OptionReg.eq_trans _ _ _ H2). -apply IHl. clear H1 H2. -intros. generalize (H0 H1). intro. -inversion H2; subst. elim n. auto. -auto. -inversion HNoDup;auto. -auto. -Qed. - -Lemma proper_coloring_empty : forall g palette, -VertexSet.Subset (precolored g) palette -> -proper_coloring (precoloring g) g palette. - -Proof. -intros g palette H. -unfold proper_coloring;unfold proper_coloring_1; -unfold proper_coloring_2;unfold proper_coloring_3. -split;[|split]. -intros e x y H0 H1 H2 H3 H4. -destruct (proper_coloring_empty_aux g (fst_ext e)); -generalize (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H2) H5); -clear H2 H5;intro H2. -destruct (proper_coloring_empty_aux g (snd_ext e)); -generalize (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H3) H5); -clear H3 H5;intro H3. -elim (In_graph_edge_diff_ext _ _ H1). -inversion H2;inversion H3;subst. -apply (Register.eq_trans (Register.eq_sym H10) (Register.eq_trans (Register.eq_sym H4) H7)). -inversion H3. -inversion H2. -intros x y H0. -assert (VertexSet.In x (precolored g)). -apply in_empty_coloring_in_precolored with (y := y). assumption. -rewrite precolored_equiv in H1. intuition. -intros x y H0. -apply H. -destruct (proper_coloring_empty_aux g x); -generalize (OptionReg.eq_trans _ _ _ (OptionReg.eq_sym _ _ H1) H0); -intro H2;inversion H2;subst. -rewrite <-H5. -apply in_empty_coloring_in_precolored with (y := x). -assumption. -Qed. - -Lemma proper_coloring_IRC_map : forall gp, -VertexSet.Subset (precolored (irc_g gp)) (pal gp) -> -proper_coloring (map_to_coloring (IRC_map gp)) (irc_g gp) (pal gp). - -Proof. -intros gp Hpre. -functional induction IRC_map gp;simpl in *. - -(* simplify *) -generalize (simplify_inv _ _ e). intro. -generalize (simplify_inv2 _ _ e). intro. destruct H0. simpl in H0. -apply available_coloring_1. -apply proper_remove_proper with (x:=r). -rewrite H0 in *. unfold simplify_irc in *. simpl in *. -apply IHt0. rewrite precolored_remove_vertex. -unfold VertexSet.Subset. intros. apply Hpre. apply (VertexSet.remove_3 H1). - -unfold compat_col. -intros. unfold map_to_coloring. -rewrite (MapFacts.find_o _ H1). apply OptionReg.eq_refl. - -apply (In_simplify_props _ _ _ _ _ _ _ _ (VertexSet.choose_1 H) (refl_equal _) (HWS_irc g)). - -(*coalesce*) -generalize (coalesce_inv _ _ e0). intro. -generalize (coalesce_inv_2 _ _ e0). intro. -destruct H0. destruct H0. simpl in *. - -assert (forall e', EdgeSet.In e' (get_movesWL (irc_wl g)) -> In_graph_edge e' (irc_g g)). -intros. generalize (In_move_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (HWS_irc g)). intuition. - -apply (complete_coloring_coloring _ _ _ x x0). -apply (any_coalescible_edge_2 e1 _ _ (get_movesWL (irc_wl g)) H1 H). -rewrite H0 in *. unfold merge_irc in *. simpl in *. apply IHt0. -rewrite precolored_merge. -unfold VertexSet.Subset. intros. apply Hpre. apply (VertexSet.remove_3 H2). -apply (proj1 (any_coalescible_edge_1 _ _ _ _ H1 H)). - -unfold compat_col. -intros. unfold map_to_coloring. -rewrite (MapFacts.find_o _ H2). apply OptionReg.eq_refl. - -(* freeze *) -generalize (freeze_inv _ _ e1). intro. -generalize (freeze_inv2 _ _ e1). intro. destruct H0. destruct H0. simpl in *. -apply (coloring_delete_preference _ r (irc_g g) x0 _). -rewrite H0 in *. unfold delete_preference_edges_irc2 in *. simpl in *. apply IHt0. -unfold VertexSet.Subset. intros. apply Hpre. -rewrite precolored_delete_preference_edges in H1. assumption. - -(* spill *) -generalize (spill_inv _ _ e2). intro. -generalize (spill_inv2 _ _ e2). intro. destruct H0. simpl in *. -apply available_coloring_1. -apply proper_remove_proper with (x:=r). -rewrite H0 in *. unfold spill_irc in *. simpl in *. apply IHt0. -unfold VertexSet.Subset. intros. apply Hpre. -rewrite precolored_remove_vertex in H1. apply (VertexSet.remove_3 H1). - -unfold compat_col. -intros. unfold map_to_coloring. -rewrite (MapFacts.find_o _ H1). apply OptionReg.eq_refl. - -apply (In_spill_props _ _ _ _ _ _ _ _ (lowest_cost_in _ _ _ H) (refl_equal _) (HWS_irc g)). - -(* terminal case *) -apply proper_coloring_empty. assumption. -Qed. - -Lemma proper_coloring_IRC_aux : forall g palette, -VertexSet.Subset (precolored g) palette -> -proper_coloring (IRC g palette) g palette. - -Proof. -intros. apply proper_coloring_IRC_map. auto. -Qed. - -Lemma proper_coloring_IRC : forall g palette, -proper_coloring (precoloring g) g palette -> -proper_coloring (IRC g palette) g palette. - -Proof. -intros. apply proper_coloring_IRC_map. -intro. intro. unfold proper_coloring, proper_coloring_3 in H. -do 2 destruct H as [_ H]. unfold graph_to_IRC_graph in *. simpl in *. -apply H with (x:= a). apply proper_coloring_pre_inv. assumption. -Qed.
\ No newline at end of file diff --git a/backend/IRC_Graph_Functions.v b/backend/IRC_Graph_Functions.v deleted file mode 100755 index fc691aa..0000000 --- a/backend/IRC_Graph_Functions.v +++ /dev/null @@ -1,548 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Spill_WL. -Require Import ZArith. -Require Import Simplify_WL. -Require Import Spill_WL. -Require Import Merge_WL. -Require Import Freeze_WL. -Require Import IRC_graph. -Require Import Edges. -Require Import Conservative_criteria. -Require Import WS. - -Import RegFacts Props OTFacts. - -Definition any_vertex := VertexSet.choose. - -(* simplify *) - -Definition simplify_irc r ircg H := -Make_IRC_Graph (remove_vertex r (irc_g ircg)) - (simplify_wl r ircg (irc_k ircg)) - (pal ircg) - (irc_k ircg) - (WS_simplify r ircg H) - (Hk ircg). - -Definition simplify g : option (Register.t * irc_graph) := -let simplifyWL := get_simplifyWL (irc_wl g) in -match any_vertex simplifyWL as v return (any_vertex simplifyWL = v -> option (Register.t * irc_graph)) with -|Some r => fun H : any_vertex simplifyWL = Some r => - Some (r, simplify_irc r g (VertexSet.choose_1 H)) -|None => fun H : any_vertex simplifyWL = None => None -end (refl_equal (any_vertex simplifyWL)). - -Lemma simplify_inv_aux : - forall g P, - match simplify g with - | Some x => - forall ( H : (any_vertex (get_simplifyWL (irc_wl g)) = Some (fst x))), - (simplify_irc (fst x) g (VertexSet.choose_1 H) = snd x) -> P - | None => - (any_vertex (get_simplifyWL (irc_wl g)) = None -> P) - end -> P. -Proof. - intros g P. - unfold simplify. - - set (simplifyWL := get_simplifyWL (irc_wl g)) in *. - set (Z := any_vertex simplifyWL) in *. - - refine - (match Z as W - return forall (H : Z = W), - -match - match W as v return (Z = v -> option (Register.t * irc_graph)) with - | Some r => - fun H : Z = Some r => Some (r, simplify_irc r g (VertexSet.choose_1 H)) - | None => fun _ : Z = None => None - end H -with -| Some x => forall H : Z = Some (fst x), simplify_irc (fst x) g (VertexSet.choose_1 H) = snd x -> P -| None => Z = None -> P -end -> P - - with - | Some x => _ - | None => _ - end _). - -simpl. intros. apply X with (H0 := H). reflexivity. -auto. -Qed. - -Lemma simplify_inv : forall g res, -simplify g = Some res -> -any_vertex (get_simplifyWL (irc_wl g)) = Some (fst res). -Proof. - intros. - apply simplify_inv_aux with g. - rewrite H. - auto. -Qed. - -Lemma simplify_inv2 : forall g res, -simplify g = Some res -> -exists H, snd res = simplify_irc (fst res) g (VertexSet.choose_1 H). - -Proof. -intros. -apply (simplify_inv_aux g). rewrite H. -simpl. intros. rewrite <-H1. -exists H0. reflexivity. -Qed. - -(* merge *) - -Definition merge_irc e ircg pin paff := -let g' := merge e (irc_g ircg) pin paff in -Make_IRC_Graph g' - (merge_wl3 e ircg g' pin paff) - (pal ircg) - (irc_k ircg) - (WS_coalesce _ _ pin paff) - (Hk ircg). - -Definition coalesce g : option (Edge.t * irc_graph) := -let movesWL := get_movesWL (irc_wl g) in -let graph := irc_g g in -let HWS := HWS_irc g in -let k := irc_k g in -match any_coalescible_edge movesWL graph k as e -return (any_coalescible_edge movesWL graph k = e -> option (Edge.t * irc_graph)) with -|Some edge => fun H : any_coalescible_edge movesWL graph k = Some edge => - let Hin := any_coalescible_edge_11 _ _ _ _ H in - let Hing := proj2 (In_move_props _ _ _ _ _ _ _ _ Hin (refl_equal _) HWS) in - let Haff := proj1 (In_move_props _ _ _ _ _ _ _ _ Hin (refl_equal _) HWS) in - Some (edge,merge_irc edge g Hing Haff) -|None => fun H : any_coalescible_edge movesWL graph k = None => None -end (refl_equal (any_coalescible_edge movesWL graph k)). - -Lemma coalesce_inv_aux : - forall g P, - match coalesce g with - | Some x => - forall (H : (any_coalescible_edge (get_movesWL (irc_wl g)) (irc_g g) (irc_k g) = Some (fst x))), - (merge_irc (fst x) g - (proj2 (In_move_props _ _ _ _ _ _ _ _ (any_coalescible_edge_11 _ _ _ _ H) (refl_equal _) (HWS_irc g))) - (proj1 (In_move_props _ _ _ _ _ _ _ _ (any_coalescible_edge_11 _ _ _ _ H) (refl_equal _) (HWS_irc g)))) - = snd x -> P - | None => - (any_coalescible_edge (get_movesWL (irc_wl g)) (irc_g g) (irc_k g) = None -> P) - end -> P. -Proof. - intros g P. - unfold coalesce. - - set (movesWL := get_movesWL (irc_wl g)) in *. - set (Z := any_coalescible_edge movesWL (irc_g g) (irc_k g)) in *. - - refine - (match Z as W - return forall (H : Z = W), - -match - match W as e return (Z = e -> option (Edge.t * irc_graph)) with - | Some edge => - fun H : Z = Some edge => Some (edge, -merge_irc edge g - (proj2 - (In_move_props edge (irc_g g) - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL) movesWL - (get_spillWL (irc_wl g)) (get_freezeWL (irc_wl g)) - (get_simplifyWL (irc_wl g)) (irc_k g) - (any_coalescible_edge_11 edge (irc_g g) (irc_k g) movesWL H) - (refl_equal - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL)) (HWS_irc g))) - (proj1 - (In_move_props edge (irc_g g) - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL) movesWL - (get_spillWL (irc_wl g)) (get_freezeWL (irc_wl g)) - (get_simplifyWL (irc_wl g)) (irc_k g) - (any_coalescible_edge_11 edge (irc_g g) (irc_k g) movesWL H) - (refl_equal - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL)) (HWS_irc g)))) - | None => fun _ : Z = None => None - end H -with -| Some x => - forall H : Z = Some (fst x), - merge_irc (fst x) g - (proj2 - (In_move_props (fst x) (irc_g g) - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL) movesWL - (get_spillWL (irc_wl g)) (get_freezeWL (irc_wl g)) - (get_simplifyWL (irc_wl g)) (irc_k g) - (any_coalescible_edge_11 (fst x) (irc_g g) (irc_k g) movesWL H) - (refl_equal - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL)) (HWS_irc g))) - (proj1 - (In_move_props (fst x) (irc_g g) - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL) movesWL - (get_spillWL (irc_wl g)) (get_freezeWL (irc_wl g)) - (get_simplifyWL (irc_wl g)) (irc_k g) - (any_coalescible_edge_11 (fst x) (irc_g g) (irc_k g) movesWL H) - (refl_equal - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), movesWL)) (HWS_irc g))) = snd x -> - P -| None => Z = None -> P -end -> P - with - | Some x => _ - | None => _ - end _). - -simpl. intros. apply X with (H0 := H). reflexivity. -auto. -Qed. - -Lemma coalesce_inv : forall g res, -coalesce g = Some res -> -any_coalescible_edge (get_movesWL (irc_wl g)) (irc_g g) (irc_k g) = Some (fst res). -Proof. - intros. - apply (coalesce_inv_aux g). - rewrite H. - auto. -Qed. - -Lemma coalesce_inv_2 : forall g res, -coalesce g = Some res -> -exists H, exists H', snd res = merge_irc (fst res) g H H'. - -Proof. -intros. -apply (coalesce_inv_aux g). -rewrite H. -simpl. intros. -exists ((proj2 - (In_move_props (fst res) (irc_g g) - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g)) - (get_movesWL (irc_wl g)) (get_spillWL (irc_wl g)) - (get_freezeWL (irc_wl g)) (get_simplifyWL (irc_wl g)) - (irc_k g) - (any_coalescible_edge_11 (fst res) (irc_g g) (irc_k g) - (get_movesWL (irc_wl g)) H0) - (refl_equal - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g))) - (HWS_irc g)))). -exists ((proj1 - (In_move_props (fst res) (irc_g g) - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g)) - (get_movesWL (irc_wl g)) (get_spillWL (irc_wl g)) - (get_freezeWL (irc_wl g)) (get_simplifyWL (irc_wl g)) - (irc_k g) - (any_coalescible_edge_11 (fst res) (irc_g g) (irc_k g) - (get_movesWL (irc_wl g)) H0) - (refl_equal - (get_spillWL (irc_wl g), get_freezeWL (irc_wl g), - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g))) - (HWS_irc g)))). -auto. -Qed. - -(* freeze *) - -Definition delete_preference_edges_irc2 v ircg Hing Hdep := -let k := irc_k ircg in -let g' := delete_preference_edges v (irc_g ircg) Hing in -Make_IRC_Graph g' - (delete_preferences_wl2 v ircg k) - (pal ircg) - (irc_k ircg) - (WS_freeze v ircg Hing Hdep) - (Hk ircg). - -Definition freeze g : option (Register.t * irc_graph) := -let freezeWL := get_freezeWL (irc_wl g) in -let graph := irc_g g in -let HWS := HWS_irc g in -match any_vertex freezeWL as r -return (any_vertex freezeWL = r -> option (Register.t*irc_graph)) with -|Some x => fun H : any_vertex freezeWL = Some x => - let Hin := VertexSet.choose_1 H in - let Hing := proj1 (proj2 (proj2 (In_freeze_props _ _ _ _ _ _ _ _ Hin (refl_equal _) HWS)))in - Some (x,delete_preference_edges_irc2 x g Hing Hin) -|None => fun H : any_vertex freezeWL = None => None -end (refl_equal (any_vertex freezeWL)). - -Lemma freeze_inv_aux : - forall g P, - match freeze g with - | Some x => - forall ( H : (any_vertex (get_freezeWL (irc_wl g)) = Some (fst x))), - (delete_preference_edges_irc2 (fst x) g - (proj1 (proj2 (proj2 (In_freeze_props _ _ _ _ _ _ _ _ (VertexSet.choose_1 H) (refl_equal _) (HWS_irc g))))) - (VertexSet.choose_1 H) = snd x) -> P - | None => - (any_vertex (get_freezeWL (irc_wl g)) = None -> P) - end -> P. - -Proof. - intros g P. - unfold freeze. - - set (freezeWL := get_freezeWL (irc_wl g)) in *. - set (Z := any_vertex freezeWL) in *. - - refine - (match Z as W - return forall (H : Z = W), - -match - match W as v return (Z = v -> option (Register.t * irc_graph)) with - | Some x => - fun H : Z = Some x => Some - (x, - delete_preference_edges_irc2 x g - (proj1 - (proj2 - (proj2 - (In_freeze_props x (irc_g g) - (get_spillWL (irc_wl g), freezeWL, - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g)) - freezeWL (get_spillWL (irc_wl g)) - (get_simplifyWL (irc_wl g)) (get_movesWL (irc_wl g)) - (irc_k g) (VertexSet.choose_1 H) - (refl_equal - (get_spillWL (irc_wl g), freezeWL, - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g))) - (HWS_irc g))))) (VertexSet.choose_1 H)) - | None => fun _ : Z = None => None - end H -with -| Some x => - -forall H : Z = Some (fst x), - delete_preference_edges_irc2 (fst x) g - (proj1 - (proj2 - (proj2 - (In_freeze_props (fst x) (irc_g g) - (get_spillWL (irc_wl g), freezeWL, - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g)) freezeWL - (get_spillWL (irc_wl g)) (get_simplifyWL (irc_wl g)) - (get_movesWL (irc_wl g)) (irc_k g) - (VertexSet.choose_1 H) - (refl_equal - (get_spillWL (irc_wl g), freezeWL, - get_simplifyWL (irc_wl g), get_movesWL (irc_wl g))) - (HWS_irc g))))) (VertexSet.choose_1 H) = snd x -> P -| None => Z = None -> P -end -> P - with - | Some x => _ - | None => _ - end _). - -simpl. intros. apply X with (H0 := H). reflexivity. -auto. -Qed. - -Lemma freeze_inv : forall g res, -freeze g = Some res -> -any_vertex (get_freezeWL (irc_wl g)) = Some (fst res). -Proof. - intros. - apply freeze_inv_aux with g. - rewrite H. - auto. -Qed. - -Lemma freeze_inv2 : forall g res, -freeze g = Some res -> -exists H', exists H, snd res = delete_preference_edges_irc2 (fst res) g H H'. - -Proof. -intros. -apply (freeze_inv_aux g). rewrite H. -simpl. intros. rewrite <-H1. -exists (VertexSet.choose_1 H0). -exists (proj1 (proj2 (proj2 (In_freeze_props _ _ _ _ _ _ _ _ (VertexSet.choose_1 H0) (refl_equal _) (HWS_irc g))))). reflexivity. -Qed. - -(* spill *) -Definition spill_irc r ircg H := -Make_IRC_Graph (remove_vertex r (irc_g ircg)) - (spill_wl r ircg (irc_k ircg)) - (pal ircg) - (irc_k ircg) - (WS_spill r ircg H) - (Hk ircg). - -Definition cost_order (opt : (Register.t*nat*nat)) y g := -let (tmp, pref_card) := opt in -let (x, int_card) := tmp in -let y_int := VertexSet.cardinal (interference_adj y g) in -match lt_eq_lt_dec y_int int_card with -|inleft (left _) => opt -|inleft (right _) => let y_pref := VertexSet.cardinal (preference_adj y g) in - match le_lt_dec pref_card y_pref with - |left _ => opt - |right _ => (y, y_int, y_pref) - end -|inright _ => (y, y_int, VertexSet.cardinal (preference_adj y g)) -end. - -Definition lowest_cost_aux s g o := -VertexSet.fold (fun v o => match o with - | Some opt => Some (cost_order opt v g) - | None => Some (v, VertexSet.cardinal (interference_adj v g), - VertexSet.cardinal (preference_adj v g)) - end) - s - o. - -Definition lowest_cost s g := -match lowest_cost_aux s g None with -| Some r => Some (fst (fst r)) -| None => None -end. - -Lemma lowest_cost_aux_in : forall x s g o, -lowest_cost_aux s g o = Some x-> -VertexSet.In (fst (fst x)) s \/ o = Some x. - -Proof. -intros. unfold lowest_cost_aux in H. -set (f := (fun (v : VertexSet.elt) (o : option (MyRegisters.Regs.t * nat * nat)) => - match o with - | Some opt => Some (cost_order opt v g) - | None => - Some - (v, VertexSet.cardinal (interference_adj v g), - VertexSet.cardinal (preference_adj v g)) - end )) in *. -unfold VertexSet.elt in *. -fold f in H. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -unfold VertexSet.elt in *. fold f' in H. -generalize VertexSet.elements_2. intro HH. -generalize (HH s). clear HH. intro HH. -generalize x o H. clear H. -induction (VertexSet.elements s). intros x0 o0 H. -simpl in H. right. auto. -simpl. intros. -assert (VertexSet.In (fst (fst x0)) s \/ (f' o0 a) = Some x0). -apply IHl. -intros. apply HH. right. auto. -auto. -destruct H0. -left. auto. -unfold f' in H0. unfold f in H0. -case_eq o0; intros. -rewrite H1 in *. -case_eq (cost_order p a g); intros. -rewrite H2 in H0. unfold cost_order in H2. -destruct p. simpl in *. destruct p. simpl in *. -destruct (lt_eq_lt_dec (VertexSet.cardinal (interference_adj a g)) n1). destruct s0. -right. rewrite H2. auto. -destruct (le_lt_dec n0 (VertexSet.cardinal (preference_adj a g))). -right. rewrite H2. auto. -left. apply HH. left. destruct p0. simpl in *. -destruct x0. destruct p. simpl in *. -inversion H0. inversion H2. subst. intuition. -left. apply HH. left. destruct p0. simpl in *. -destruct x0. destruct p. simpl in *. -inversion H0. inversion H2. subst. intuition. -rewrite H1 in H0. -unfold f' in H. rewrite H1 in H. simpl in H. rewrite H0 in H. -fold f' in H. -left. apply HH. inversion H0. simpl. left. intuition. -Qed. - -Lemma lowest_cost_in : forall x s g, -lowest_cost s g = Some x -> -VertexSet.In x s. - -Proof. -intros. unfold lowest_cost in H. -case_eq (lowest_cost_aux s g None); intros; rewrite H0 in H. -generalize (lowest_cost_aux_in p s g None H0). intro. -destruct H1. inversion H. subst. auto. -congruence. -congruence. -Qed. - -Definition spill g : option (Register.t * irc_graph) := -let spillWL := get_spillWL (irc_wl g) in -match lowest_cost spillWL (irc_g g) as v return (lowest_cost spillWL (irc_g g) = v -> option (Register.t * irc_graph)) with -|Some r => fun H : lowest_cost spillWL (irc_g g) = Some r => - Some (r, spill_irc r g (lowest_cost_in _ _ _ H)) -|None => fun H : lowest_cost spillWL (irc_g g) = None => None -end (refl_equal (lowest_cost spillWL (irc_g g))). - -Lemma spill_inv_aux : - forall g P, - match spill g with - | Some x => - forall ( H : (lowest_cost (get_spillWL (irc_wl g)) (irc_g g) = Some (fst x))), - (spill_irc (fst x) g (lowest_cost_in _ _ _ H) = snd x) -> P - | None => - (lowest_cost (get_spillWL (irc_wl g)) (irc_g g) = None -> P) - end -> P. -Proof. - intros g P. - unfold spill. - - set (spillWL := get_spillWL (irc_wl g)) in *. - set (Z := lowest_cost spillWL (irc_g g)) in *. - - refine - (match Z as W - return forall (H : Z = W), - -match - match W as v return (Z = v -> option (Register.t * irc_graph)) with - | Some r => - fun H : Z = Some r => Some (r, spill_irc r g (lowest_cost_in _ _ _ H)) - | None => fun _ : Z = None => None - end H -with -| Some x => forall H : Z = Some (fst x), spill_irc (fst x) g (lowest_cost_in _ _ _ H) = snd x -> P -| None => Z = None -> P -end -> P - - with - | Some x => _ - | None => _ - end _). - -simpl. intros. apply X with (H0 := H). reflexivity. -auto. -Qed. - -Lemma spill_inv : forall g res, -spill g = Some res -> -lowest_cost (get_spillWL (irc_wl g)) (irc_g g) = Some (fst res). -Proof. - intros. - apply spill_inv_aux with g. - rewrite H. - auto. -Qed. - -Lemma spill_inv2 : forall g res, -spill g = Some res -> -exists H, snd res = spill_irc (fst res) g (lowest_cost_in (fst res) (get_spillWL (irc_wl g)) (irc_g g) H). - -Proof. -intros. -apply (spill_inv_aux g). rewrite H. -simpl. intros. rewrite <-H1. -exists H0. reflexivity. -Qed. diff --git a/backend/IRC_graph.v b/backend/IRC_graph.v deleted file mode 100755 index 31e9ce8..0000000 --- a/backend/IRC_graph.v +++ /dev/null @@ -1,15 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. - -Record irc_graph := Make_IRC_Graph { -irc_g : tt; -irc_wl : WS; -pal : VertexSet.t; -irc_k : nat; -HWS_irc : WS_properties irc_g irc_k irc_wl; -Hk : VertexSet.cardinal pal = irc_k -}. - -Definition graph_to_IRC_graph g palette := -let K := VertexSet.cardinal palette in -Make_IRC_Graph g (get_WL g K) palette K (WS_prop_get _ _) (refl_equal _). diff --git a/backend/IRC_termination.v b/backend/IRC_termination.v deleted file mode 100755 index f2f7d9f..0000000 --- a/backend/IRC_termination.v +++ /dev/null @@ -1,397 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import IRC_Graph_Functions. -Require Import ZArith. -Require Import IRC_Graph_Functions. -Require Import WS. -Require Import IRC_graph. -Require Import Simplify_WL. -Require Import Spill_WL. -Require Import Freeze_WL. -Require Import Merge_WL. -Require Import MyRegisters. -Require Import Remove_Vertex_WL. -Require Import Affinity_relation. -Require Import Edges. -Require Import Merge_Move. -Require Import Merge_Degree. -Require Import Conservative_criteria. -Require Import Delete_Preference_Edges_Degree. -Require Import Delete_Preference_Edges_Move. -Require Import Remove_Vertex_Move. -Require Import Remove_Vertex_Degree. - -Import Edge RegFacts Props OTFacts. - -Definition not_precolored_irc g := not_precolored (irc_g g). - -Definition irc_measure g := -let simplifyWL := get_simplifyWL (irc_wl g) in -let m := VertexSet.cardinal simplifyWL in -let np := not_precolored_irc g in -let n := VertexSet.cardinal np in -2*n - m. - -Lemma empty_union_empty : -VertexSet.Equal -(VertexSet.union VertexSet.empty VertexSet.empty) -(VertexSet.empty). - -Proof. -split;intros. -destruct (VertexSet.union_1 H); elim (VertexSet.empty_1 H0). -elim (VertexSet.empty_1 H). -Qed. - -Lemma cardinal_union_not_precolored : forall g, -VertexSet.cardinal (not_precolored_irc g) = -VertexSet.cardinal (get_simplifyWL (irc_wl g)) + -VertexSet.cardinal (get_freezeWL (irc_wl g)) + -VertexSet.cardinal (get_spillWL (irc_wl g)). - -Proof. -intros. unfold not_precolored_irc. rewrite <-(not_precolored_union_ws _ _ (irc_wl g)). -do 2 rewrite union_cardinal_inter. rewrite union_inter_1. -generalize (WS_empty_inter_1 _ _ (irc_wl g) (HWS_irc g)). intro. -generalize (WS_empty_inter_2 _ _ (irc_wl g) (HWS_irc g)). intro. -generalize (WS_empty_inter_3 _ _ (irc_wl g) (HWS_irc g)). intro. -generalize (empty_is_empty_1 H). intro. -generalize (empty_is_empty_1 H0). intro. -generalize (empty_is_empty_1 H1). intro. -rewrite H2. rewrite H3. rewrite H4. rewrite empty_union_empty. -rewrite empty_cardinal. omega. -apply HWS_irc. -Qed. - -Lemma irc_measure_equiv_aux : forall g, -irc_measure g = VertexSet.cardinal (get_simplifyWL (irc_wl g)) + - VertexSet.cardinal (get_freezeWL (irc_wl g)) + - VertexSet.cardinal (get_freezeWL (irc_wl g)) + - VertexSet.cardinal (get_spillWL (irc_wl g)) + - VertexSet.cardinal (get_spillWL (irc_wl g)). - -Proof. -intros. unfold irc_measure. simpl. -rewrite cardinal_union_not_precolored. repeat rewrite <-plus_n_O. omega. -Qed. - -Definition card_n g := VertexSet.cardinal (not_precolored_irc g). -Definition card_p g := -VertexSet.cardinal (VertexSet.union (get_freezeWL (irc_wl g)) (get_spillWL (irc_wl g))). - -Lemma irc_measure_equiv : forall g, -irc_measure g = card_n g + card_p g. - -Proof. -intros. unfold card_n, card_p. -rewrite irc_measure_equiv_aux. rewrite cardinal_union_not_precolored. -repeat rewrite <-plus_assoc. -rewrite plus_comm. rewrite <-plus_assoc. -rewrite union_cardinal. omega. -intros. intro. destruct H. elim (WS_empty_inter_1 _ _ _ (HWS_irc g) x). -apply VertexSet.inter_3; auto. -Qed. - -Lemma remove_cardinal_weak_dec : forall x s, -VertexSet.cardinal s - 1 <= VertexSet.cardinal (VertexSet.remove x s). - -Proof. -intros. -destruct (In_dec x s); intros. -rewrite <-(remove_cardinal_1 i). omega. -rewrite <-(remove_cardinal_2 n). omega. -Qed. - -Lemma simplify_dec_aux_1 : forall g r H, -card_n (simplify_irc r g H) < card_n g. - -Proof. -intros. unfold card_n, simplify_irc, not_precolored_irc, not_precolored; simpl. -generalize (In_simplify_props _ _ _ _ _ _ _ _ H (refl_equal _) (HWS_irc g)). intro. -destruct H0. destruct H1. destruct H2. -apply subset_cardinal_lt with (x:=r). -unfold VertexSet.Subset. intros. -generalize (VertexSet.diff_1 H4). intro. -generalize (VertexSet.diff_2 H4). clear H4. intro. -apply VertexSet.diff_3. apply (VertexSet.remove_3 H5). -rewrite precolored_remove_vertex in H4. intro. elim H4. -apply VertexSet.remove_2; auto. -intro. elim (VertexSet.remove_1 H7 H5). -apply VertexSet.diff_3; auto. -intro. generalize (VertexSet.diff_1 H4). intro. - generalize (VertexSet.diff_2 H4). intro. -elim (VertexSet.remove_1 (MyRegisters.Regs.eq_refl _) H5). -Qed. - -Lemma simplify_dec_aux_2 : forall g r H, -card_p (simplify_irc r g H) <= card_p g. - -Proof. -intros. unfold card_p, simplify_irc, not_precolored_irc, not_precolored; simpl. -apply subset_cardinal. unfold simplify_wl. -case_eq (VertexSet.partition (move_related (irc_g g)) - (VertexSet.filter - (fun x : VertexSet.elt => - eq_K (irc_k g) - (VertexSet.cardinal (interference_adj x (irc_g g)))) - (VertexSet.diff (interference_adj r (irc_g g)) - (precolored (irc_g g))))); intros. -unfold get_spillWL, get_freezeWL; simpl. -assert (VertexSet.Equal t0 (fst (VertexSet.partition (move_related (irc_g g)) - (VertexSet.filter - (fun x : VertexSet.elt => - eq_K (irc_k g) - (VertexSet.cardinal (interference_adj x (irc_g g)))) - (VertexSet.diff (interference_adj r (irc_g g)) - (precolored (irc_g g))))))) as Ht0. -rewrite H0. simpl. apply VertexSet.eq_refl. -rewrite VertexSet.partition_1 in Ht0. - -unfold VertexSet.Subset; intros. -destruct (VertexSet.union_1 H1). -destruct (VertexSet.union_1 H2). -apply VertexSet.union_2. auto. -apply VertexSet.union_3. WS_apply (HWS_irc g). rewrite Ht0 in H3. clear H0 H1. -generalize (VertexSet.filter_1 (compat_bool_move _ ) H3). intro. -generalize (VertexSet.filter_2 (compat_bool_move _) H3). clear H3. intro. -generalize (VertexSet.filter_1 (eq_K_compat _ _) H0). intro. -generalize (VertexSet.filter_2 (eq_K_compat _ _) H0). clear H0. intro. -generalize (eq_K_2 _ _ H0). clear H0. intro. -generalize (VertexSet.diff_1 H3). intro. -generalize (VertexSet.diff_2 H3). clear H3. intro. -split. -unfold has_low_degree, interf_degree. rewrite <-H0. -destruct (le_lt_dec (irc_k g) (irc_k g)). reflexivity. elim (lt_irrefl _ l). -split. -apply move_related_in_graph; auto. -assumption. -apply VertexSet.union_3. apply (VertexSet.diff_1 H2). -apply compat_bool_move. -Qed. - -Lemma simplify_dec : forall (g : irc_graph) (r : Register.t) (g' : irc_graph), -simplify g = Some (r, g') -> irc_measure g' < irc_measure g. - -Proof. -intros. do 2 rewrite irc_measure_equiv. -generalize (simplify_inv _ _ H). intro. -generalize (simplify_inv2 _ _ H). intro. simpl in *. -destruct H1 as [H1 H2]. clear H H0. rewrite H2 in *. -generalize (simplify_dec_aux_1 g r (VertexSet.choose_1 H1)). -generalize (simplify_dec_aux_2 g r (VertexSet.choose_1 H1)). omega. -Qed. - -Lemma coalesce_dec_aux_1 : forall g e p q, -~VertexSet.In (snd_ext e) (precolored (irc_g g)) -> -card_n (merge_irc e g p q) < card_n g. - -Proof. -intros. unfold card_n, merge_irc, not_precolored_irc, not_precolored; simpl. -apply subset_cardinal_lt with (x:=(snd_ext e)). -unfold VertexSet.Subset. intros. -generalize (VertexSet.diff_1 H0). intro. -generalize (VertexSet.diff_2 H0). clear H0. intro. -apply VertexSet.diff_3. -apply (VertexSet.remove_3 H1). -rewrite precolored_merge in H0. intro. elim H0. -apply VertexSet.remove_2; auto. -intro. elim (VertexSet.remove_1 H3 H1). -apply VertexSet.diff_3. apply (proj2 (In_graph_edge_in_ext _ _ p)). auto. -intro. generalize (VertexSet.diff_1 H0). intro. - generalize (VertexSet.diff_2 H0). intro. -elim (VertexSet.remove_1 (Regs.eq_refl _) H1). -Qed. - -Lemma coalesce_dec_aux_2 : forall g e p q, -~VertexSet.In (snd_ext e) (precolored (irc_g g)) -> -card_p (merge_irc e g p q) <= card_p g. - -Proof. -intros. unfold card_p, merge_irc, not_precolored_irc, not_precolored; simpl. -apply subset_cardinal. unfold VertexSet.Subset; intros. -destruct (VertexSet.union_1 H0). -generalize (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (ws_merge3 _ _ _ _)). intro. -destruct H2. destruct H3. destruct H4. -case_eq (has_low_degree (irc_g g) (irc_k g) a); intros. -apply VertexSet.union_2. WS_apply (HWS_irc g). -split. assumption. -split. eapply move_related_merge_move_related; eauto. -intro. elim H5. rewrite precolored_merge. apply VertexSet.remove_2. -rewrite In_merge_vertex in H4. destruct H4. auto. -assumption. -apply VertexSet.union_3. WS_apply (HWS_irc g). -split. assumption. -split. rewrite In_merge_vertex in H4. destruct H4. auto. -intro. elim H5. rewrite precolored_merge. apply VertexSet.remove_2. -rewrite In_merge_vertex in H4. destruct H4. auto. -assumption. - -generalize (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (ws_merge3 _ _ _ _)). intro. -destruct H2. destruct H3. -rewrite In_merge_vertex in H3. destruct H3. -destruct (Register.eq_dec a (fst_ext e)). -rewrite e0. case_eq (has_low_degree (irc_g g) (irc_k g) (fst_ext e)); intros. -apply VertexSet.union_2. WS_apply (HWS_irc g). -split. assumption. -split. apply (proj1 (Aff_edge_aff _ _ p q)). rewrite e0 in H4. -intro. elim H4. rewrite precolored_merge. apply VertexSet.remove_2. -apply (In_graph_edge_diff_ext _ _ p). -assumption. -apply VertexSet.union_3. WS_apply (HWS_irc g). -split. assumption. -split. rewrite <-e0. assumption. -rewrite e0 in H4. -intro. elim H4. rewrite precolored_merge. apply VertexSet.remove_2. -apply (In_graph_edge_diff_ext _ _ p). -assumption. - -apply VertexSet.union_3. WS_apply (HWS_irc g). -split. rewrite Hk in H2. eapply merge_low_1; eauto. -split. assumption. -intro. elim H4. rewrite precolored_merge. apply VertexSet.remove_2; auto. -Qed. - -Lemma coalesce_dec : forall (g : irc_graph) (e : Edge.t) (g' : irc_graph), -coalesce g = Some (e, g') -> irc_measure g' < irc_measure g. - -Proof. -intros. do 2 rewrite irc_measure_equiv. -generalize (coalesce_inv _ _ H). intro. -generalize (coalesce_inv_2 _ _ H). intro. simpl in *. -destruct H1 as [H1 H2]. destruct H2 as [H2 H3]. clear H. rewrite H3 in *. -cut (forall e', EdgeSet.In e' (get_movesWL (irc_wl g)) -> In_graph_edge e' (irc_g g)). intro HH. -generalize (any_coalescible_edge_1 _ _ _ _ HH H0). intro. destruct H. -generalize (any_coalescible_edge_2 _ _ _ _ HH H0). intro. -generalize (coalesce_dec_aux_1 g e H1 H2 H5). -generalize (coalesce_dec_aux_2 g e H1 H2 H5). omega. -intros. apply (proj2(In_move_props _ _ _ _ _ _ _ _ H (refl_equal _) (HWS_irc g))). -Qed. - -Lemma freeze_dec_aux_1 : forall g x H H0, -card_n (delete_preference_edges_irc2 x g H H0) <= card_n g. - -Proof. -intros. unfold card_n, delete_preference_edges_irc2, not_precolored_irc, not_precolored; simpl. -apply subset_cardinal. unfold VertexSet.Subset; intros. -generalize (VertexSet.diff_1 H1). intro. -generalize (VertexSet.diff_2 H1). clear H1. intro. -rewrite precolored_delete_preference_edges in H1. -apply VertexSet.diff_3; assumption. -Qed. - -Lemma freeze_dec_aux_2 : forall g x H H0, -card_p (delete_preference_edges_irc2 x g H H0) < card_p g. - -Proof. -intros. unfold card_p, delete_preference_edges_irc2, not_precolored_irc, not_precolored; simpl. -generalize (In_freeze_props _ _ _ _ _ _ _ _ H0 (refl_equal _) (HWS_irc g)). intro. -destruct H1. destruct H2. destruct H3. -apply subset_cardinal_lt with (x:=x). unfold VertexSet.Subset; intros. -destruct (VertexSet.union_1 H5). -apply VertexSet.union_2. WS_apply (HWS_irc g). -generalize (In_freeze_props _ _ _ _ _ _ _ _ H6 (refl_equal _) (WS_freeze _ _ H H0)). intro. -destruct H7. destruct H8. destruct H9. -split. -rewrite <-delete_preference_edges_low in H7. auto. -split. eapply move_related_delete_move_related; eauto. -rewrite precolored_delete_preference_edges in H10. assumption. -apply VertexSet.union_3. WS_apply (HWS_irc g). -generalize (In_spill_props _ _ _ _ _ _ _ _ H6 (refl_equal _) (WS_freeze _ _ H H0)). intro. -destruct H7. destruct H8. -split. -rewrite <-delete_preference_edges_low in H7. auto. -split. rewrite In_delete_preference_edges_vertex in H8. auto. -rewrite precolored_delete_preference_edges in H9. assumption. - -apply VertexSet.union_2. assumption. - -intro. -destruct (VertexSet.union_1 H5). -generalize (In_freeze_props _ _ _ _ _ _ _ _ H6 (refl_equal _) (WS_freeze _ _ H H0)). intro. -destruct H7. destruct H8. -rewrite (not_aff_related_delete_preference_edges _ _ H) in H8. congruence. -generalize (In_spill_props _ _ _ _ _ _ _ _ H6 (refl_equal _) (WS_freeze _ _ H H0)). intro. -destruct H7. -rewrite <-delete_preference_edges_low in H7. congruence. -Qed. - -Lemma freeze_dec : forall (g : irc_graph) (r : Register.t) (g' : irc_graph), -freeze g = Some (r, g') -> irc_measure g' < irc_measure g. - -Proof. -intros. do 2 rewrite irc_measure_equiv. -generalize (freeze_inv _ _ H). intro. -generalize (freeze_inv2 _ _ H). intro. simpl in *. -destruct H1 as [H1 H2]. destruct H2 as [H2 H3]. clear H. rewrite H3 in *. -generalize (freeze_dec_aux_1 g r H2 H1). -generalize (freeze_dec_aux_2 g r H2 H1). omega. -Qed. - -Lemma spill_dec_aux_1 : forall g r H, -card_n (spill_irc r g H) < card_n g. - -Proof. -intros. unfold card_n, spill_irc, not_precolored_irc, not_precolored; simpl. -generalize (In_spill_props _ _ _ _ _ _ _ _ H (refl_equal _) (HWS_irc g)). intro. -destruct H0. destruct H1. generalize H2. intro H3. -apply subset_cardinal_lt with (x:=r). -unfold VertexSet.Subset. intros. -generalize (VertexSet.diff_1 H4). intro. -generalize (VertexSet.diff_2 H4). clear H4. intro. -apply VertexSet.diff_3. -apply (VertexSet.remove_3 H5). -rewrite precolored_remove_vertex in H4. intro. elim H4. -apply VertexSet.remove_2; auto. -intro. elim (VertexSet.remove_1 H7 H5). -apply VertexSet.diff_3; auto. -intro. generalize (VertexSet.diff_1 H4). intro. - generalize (VertexSet.diff_2 H4). intro. -elim (VertexSet.remove_1 (Regs.eq_refl _) H5). -Qed. - -Lemma spill_dec_aux_2 : forall g r H, -card_p (spill_irc r g H) <= card_p g. - -Proof. -intros. unfold card_p, spill_irc, not_precolored_irc, not_precolored; simpl. -apply subset_cardinal. unfold VertexSet.Subset; intros. -destruct (VertexSet.union_1 H0). -case_eq (has_low_degree (irc_g g) (irc_k g) a); intros. -apply VertexSet.union_2. WS_apply (HWS_irc g). -generalize (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (WS_spill _ _ H)). intro. -destruct H3. destruct H4. destruct H5. -split. assumption. -split. apply move_remove_2 with (r:=r). assumption. -intro. elim H6. rewrite precolored_remove_vertex. -apply VertexSet.remove_2. rewrite In_remove_vertex in H5. destruct H5. auto. -assumption. -apply VertexSet.union_3. WS_apply (HWS_irc g). -generalize (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (WS_spill _ _ H)). intro. -destruct H3. destruct H4. destruct H5. -split. assumption. -split. rewrite In_remove_vertex in H5. destruct H5. auto. -intro. elim H6. rewrite precolored_remove_vertex. -apply VertexSet.remove_2. rewrite In_remove_vertex in H5. destruct H5. auto. -assumption. -apply VertexSet.union_3. -WS_apply (HWS_irc g). -generalize (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (WS_spill _ _ H)). intro. -destruct H2. destruct H3. -rewrite In_remove_vertex in H3. destruct H3. -split. apply not_low_remove_not_low with (r:=r); assumption. -split. assumption. -intro. elim H4. rewrite precolored_remove_vertex. -apply VertexSet.remove_2. auto. assumption. -Qed. - -Lemma spill_dec : forall (g : irc_graph) (r : Register.t) (g' : irc_graph), -spill g = Some (r, g') -> irc_measure g' < irc_measure g. - -Proof. -intros. do 2 rewrite irc_measure_equiv. -generalize (spill_inv _ _ H). intro. -generalize (spill_inv2 _ _ H). intro. simpl in *. -destruct H1 as [H1 H2]. clear H H0. rewrite H2 in *. -generalize (spill_dec_aux_1 g r (lowest_cost_in _ _ _ H1)). -generalize (spill_dec_aux_2 g r (lowest_cost_in _ _ _ H1)). omega. -Qed. diff --git a/backend/InterfGraph.v b/backend/InterfGraph.v index 081ef59..8a9dda6 100644 --- a/backend/InterfGraph.v +++ b/backend/InterfGraph.v @@ -19,11 +19,6 @@ Require Import Maps. Require Import Ordered. Require Import Registers. Require Import Locations. -Require Import AST. -Require Import Op. -Require Import RTLtyping. -Require Import RTL. -Require Import Conventions. (** Interference graphs are undirected graphs with two kinds of nodes: - RTL pseudo-registers; @@ -303,3 +298,4 @@ Proof. intros. unfold all_interf_regs. apply in_setregreg_fold'. eapply in_setregmreg_fold. eauto. Qed. + diff --git a/backend/InterfGraphMapImp.v b/backend/InterfGraphMapImp.v deleted file mode 100755 index cdd8327..0000000 --- a/backend/InterfGraphMapImp.v +++ /dev/null @@ -1,9401 +0,0 @@ -Require Import FSets. -Require Import Recdef. -Require Import ZArith. -Require Import Coq.Init.Wf. -Require Import FSetInterface. -Require Import SetsFacts. -Require Import FMaps. -Require Import OrderedOption. -Require Import FMapAVL. -Require Import Edges. -Require Import MyRegisters. - -Module Register := Regs. - -Import Edge. - -Module VertexSet := FSetAVL.Make Vertex. -Module EdgeSet := FSetAVL.Make Edge. -Module VertexMap := FMapAVL.Make Vertex. -Module MapFacts := Facts VertexMap. -Module RegFacts := MyFacts VertexSet. -Module MEdgeFacts := MyFacts EdgeSet. -Module RegRegProps := MEdgeFacts.Props. -Module Props := RegFacts.Props. - -Definition adj_set x m := -match (VertexMap.find x m) with -| None => VertexSet.empty -| Some x => x -end. - -Record tt : Type := Make_Graph { -vertices : VertexSet.t; -imap : VertexMap.t VertexSet.t; -pmap : VertexMap.t VertexSet.t; -extremities_imap : forall x, VertexMap.In x imap <-> VertexSet.In x vertices; -extremities_pmap : forall x, VertexMap.In x pmap <-> VertexSet.In x vertices; -simple_graph : forall x y, VertexSet.In x (adj_set y imap) /\ - VertexSet.In x (adj_set y pmap) -> False; -sym_imap : forall x y, VertexSet.In x (adj_set y imap) -> - VertexSet.In y (adj_set x imap); -sym_pmap : forall x y, VertexSet.In x (adj_set y pmap) -> - VertexSet.In y (adj_set x pmap); -not_eq_extremities : forall x y, VertexSet.In x (adj_set y imap) \/ - VertexSet.In x (adj_set y pmap) -> - ~Vertex.eq x y -}. - -Definition t := tt. - -Definition V := vertices. - -Definition edgemap_to_edgeset map w := -VertexMap.fold - (fun y imapy s => VertexSet.fold - (fun z s' => EdgeSet.add (y,z,w) s') - imapy - s) - map - EdgeSet.empty. - -Definition AE g := edgemap_to_edgeset (pmap g) (Some N0). - -Definition IE g := edgemap_to_edgeset (imap g) None. - -Definition is_precolored v (g : t) := is_mreg v. - -Definition imap_remove m x := -VertexMap.remove x -(VertexSet.fold - (fun y => VertexMap.add y (VertexSet.remove x (adj_set y m))) - (adj_set x m) - m -). - -Lemma change_fst : forall x y z, -fst_ext (x,y,z) = x. - -Proof. -auto. -Qed. - -Lemma change_snd : forall x y z, -snd_ext (x,y,z) = y. - -Proof. -auto. -Qed. - -Lemma change_weight : forall x y z, -get_weight (x,y,z) = z. - -Proof. -auto. -Qed. - -(* rewriting tactic *) - -Ltac change_rewrite := -repeat (try rewrite change_fst in *;try rewrite change_snd in *;try rewrite change_weight in *). - -(* two tactics for proving equality of edges *) - -Ltac Eq_eq := -apply eq_ordered_eq;unfold E.eq;split;[simpl;split;intuition|simpl;apply OptionN_as_OT.eq_refl]. - -Ltac Eq_comm_eq := rewrite edge_comm;Eq_eq. - -Lemma eq_charac : forall x y, -eq x y -> (Vertex.eq (fst_ext x) (fst_ext y) /\ Vertex.eq (snd_ext x) (snd_ext y)) \/ - (Vertex.eq (fst_ext x) (snd_ext y) /\ Vertex.eq (snd_ext x) (fst_ext y)). - -Proof. -intros x y H;unfold eq in H;unfold ordered_edge in H; -unfold get_edge in H. -destruct (OTFacts.lt_dec (snd_ext x) (fst_ext x)); -destruct (OTFacts.lt_dec (snd_ext y) (fst_ext y)); -unfold E.eq in H;simpl in H;intuition. -Qed. - -Section fold_assoc_interf_map. - -Variable A : Type. - -Inductive eq_set_option : option VertexSet.t -> option VertexSet.t -> Prop := -|None_eq : eq_set_option None None -|Some_eq : forall s s', VertexSet.Equal s s' -> eq_set_option (Some s) (Some s'). - -Definition EqualSetMap m1 m2 := forall x, eq_set_option (VertexMap.find x m1) (VertexMap.find x m2). - -Lemma EqualSetMap_refl : forall m, EqualSetMap m m. - -Proof. -unfold EqualSetMap. intro m. intro x. -destruct (VertexMap.find x m). -constructor. intuition. -constructor. -Qed. - -Lemma EqualSetMap_trans : forall m1 m2 m3, -EqualSetMap m1 m2 -> -EqualSetMap m2 m3 -> -EqualSetMap m1 m3. - -Proof. -intros m1 m2 m3 H H0. -unfold EqualSetMap in *. -intro x. -generalize (H x). clear H. intro H. -generalize (H0 x). clear H0. intro H0. -destruct (VertexMap.find x m1). -inversion H. subst. -rewrite <-H2 in H0. -destruct (VertexMap.find x m3). -constructor. inversion H0. subst. -rewrite H3. assumption. -inversion H0. -destruct (VertexMap.find x m3). -inversion H0. inversion H. subst. rewrite <-H4 in H1. inversion H1. -constructor. -Qed. - -Lemma fold_left_compat_map : forall (f : VertexMap.t VertexSet.t -> A -> VertexMap.t VertexSet.t) l e e', -EqualSetMap e e' -> -(forall e1 e2 a, EqualSetMap e1 e2 -> EqualSetMap (f e1 a) (f e2 a)) -> -EqualSetMap (fold_left f l e) (fold_left f l e'). - -Proof. -intro f;induction l;simpl. -auto. -intros e e' H H0 H1. -apply (IHl (f e a) (f e' a)). -apply H0;assumption. -assumption. -Qed. - -Lemma fold_left_assoc_map : forall l (f : VertexMap.t VertexSet.t -> A -> VertexMap.t VertexSet.t) x h, -(forall (y z : A) s, EqualSetMap (f (f s y) z) (f (f s z) y)) -> -(forall e1 e2 a, EqualSetMap e1 e2 -> EqualSetMap (f e1 a) (f e2 a)) -> -EqualSetMap (fold_left f (h :: l) x) (f (fold_left f l x) h). - -Proof. -induction l;simpl;intros f x h H H0. -apply EqualSetMap_refl. -apply EqualSetMap_trans with (m2 := fold_left f (h :: l) (f x a)). -simpl. apply fold_left_compat_map. apply H. -assumption. -apply IHl. assumption. assumption. -Qed. - -End fold_assoc_interf_map. - -Lemma fold_assoc : forall g g' y0 z s, -(forall x y a, EdgeSet.Equal (g x (g' y a)) (g' y (g x a))) -> -(forall (y z0 : VertexSet.elt) (s0 : EdgeSet.t), -EdgeSet.Equal (g z0 (g y s0)) (g y (g z0 s0))) -> -(forall (e1 e2 : EdgeSet.t) (a1 : VertexSet.elt), -EdgeSet.Equal e1 e2 -> EdgeSet.Equal (g a1 e1) (g a1 e2)) -> -(forall (e1 e2 : EdgeSet.t) (a1 : VertexSet.elt), -EdgeSet.Equal e1 e2 -> EdgeSet.Equal (g' a1 e1) (g' a1 e2)) -> -(forall (y z0 : VertexSet.elt) (s0 : EdgeSet.t), -EdgeSet.Equal (g' z0 (g' y s0)) (g' y (g' z0 s0))) -> -EdgeSet.Equal (VertexSet.fold g z (VertexSet.fold g' y0 s)) - (VertexSet.fold g' y0 (VertexSet.fold g z s)). - -Proof. -intros. -repeat rewrite VertexSet.fold_1. -set (f1 := fun (a : EdgeSet.t) (e : VertexSet.elt) => g e a). -set (f2 := fun (a : EdgeSet.t) (e : VertexSet.elt) => g' e a). -induction (VertexSet.elements z). simpl. -apply EdgeSet.eq_refl. - -set (l' := VertexSet.elements y0) in *. -assert (EdgeSet.Equal (fold_left f2 l' (fold_left f1 (a :: l) s)) - (fold_left f2 l' (f1 (fold_left f1 l s) a))). -apply MEdgeFacts.fold_left_compat_set. -apply MEdgeFacts.fold_left_assoc. - -unfold f2. assumption. -unfold f2. assumption. -unfold f1. assumption. - -apply EdgeSet.eq_trans with (s' := (fold_left f2 l' (f1 (fold_left f1 l s) a))). -set (s' := fold_left f1 l s) in *. -cut (EdgeSet.Equal (fold_left f2 l' (f1 s' a)) (f1 (fold_left f2 l' s') a)). -intro. -apply EdgeSet.eq_trans with (s' := f1 (fold_left f2 l' s') a). -rewrite MEdgeFacts.fold_left_assoc. -apply H1. assumption. -assumption. -assumption. -apply EdgeSet.eq_sym. auto. - -clear IHl. clear H4. -induction l'. simpl. apply EdgeSet.eq_refl. -assert (EdgeSet.Equal (f1 (fold_left f2 (a0 :: l') s') a) - (f1 (f2 (fold_left f2 l' s') a0) a)). -apply H1. -apply MEdgeFacts.fold_left_assoc. -assumption. -assumption. -apply EdgeSet.eq_trans with (s':= (f1 (f2 (fold_left f2 l' s') a0) a)). -rewrite MEdgeFacts.fold_left_assoc. -apply EdgeSet.eq_trans with (s' := f2 (f1 (fold_left f2 l' s') a) a0). -apply H2. assumption. -unfold f1. unfold f2. -unfold EdgeSet.eq. apply EdgeSet.eq_sym. -apply H. -assumption. -assumption. -apply H1. apply EdgeSet.eq_sym. apply MEdgeFacts.fold_left_assoc. -assumption. -assumption. -apply EdgeSet.eq_sym. auto. -Qed. - -Lemma edgemap_to_edgeset_charac : forall m x y (w : option N), -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -(EdgeSet.In (x,y,w) (edgemap_to_edgeset m w) <-> VertexSet.In y (adj_set x m)). - -Proof. -intros m x y w Hsym. split; intros. -unfold edgemap_to_edgeset in H. -rewrite VertexMap.fold_1 in H. -generalize VertexMap.elements_2. intro. -generalize (H0 _ m). clear H0. intro HH. -induction (VertexMap.elements m). -simpl in H. -elim (EdgeSet.empty_1 H). -set (f := (fun (a : EdgeSet.t) (p : VertexMap.key * VertexSet.t) => - VertexSet.fold - (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst p, z, w) s') (snd p) a)) in *. -case_eq a; intros; subst. -rewrite MEdgeFacts.fold_left_assoc in H. -set (s := fold_left f l EdgeSet.empty) in *. -unfold f in H. simpl in H. -assert (EdgeSet.In (x,y,w) s \/ (VertexSet.In y t0 /\ Vertex.eq k x) \/ - (VertexSet.In x t0 /\ Vertex.eq k y)). -clear IHl. intros. -rewrite VertexSet.fold_1 in H. -generalize VertexSet.elements_2. -intro H0. generalize (H0 t0). clear H0. intro Helt. -induction (VertexSet.elements t0). -simpl in H. left. assumption. -rewrite MEdgeFacts.fold_left_assoc in H. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H). -fold (eq (k,a,w) (x,y,w)) in H0. -right. -destruct (eq_charac _ _ H0); destruct H1; change_rewrite. -left. split. -apply Helt. left. apply Vertex.eq_sym. assumption. -assumption. -right. split. -apply Helt. left. apply Vertex.eq_sym. assumption. -assumption. -apply IHl0. assumption. -intros. apply Helt. right. assumption. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -destruct H0. -apply IHl. -assumption. -intros. apply HH. auto. -destruct H0. -assert (VertexMap.MapsTo x t0 m). -apply HH. -left. -constructor; simpl; intuition. -generalize (VertexMap.find_1 H1). clear H1. intro H1. -unfold adj_set. rewrite H1. intuition. -apply Hsym. -assert (VertexMap.MapsTo y t0 m). -apply HH. -left. -constructor; simpl; intuition. -generalize (VertexMap.find_1 H1). clear H1. intro H1. -unfold adj_set. rewrite H1. intuition. - -unfold f. -intros. set (g := (fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst z, z0, w) s')). -set (g' := fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst y0, z0, w) s'). -apply fold_assoc. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. -unfold f. -rewrite VertexSet.fold_1. -rewrite VertexSet.fold_1. -apply MEdgeFacts.fold_left_compat_set. -assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -unfold edgemap_to_edgeset. -rewrite VertexMap.fold_1. -generalize VertexMap.elements_1. intro. -case_eq (VertexMap.find x m); intros. -generalize (H0 _ m x t0). clear H0. intro HH. -induction (VertexMap.elements m). -simpl. -assert (VertexMap.MapsTo x t0 m). -apply VertexMap.find_2. assumption. -generalize (HH H0). intro H2. inversion H2. - -set (f := (fun (a : EdgeSet.t) (p : VertexMap.key * VertexSet.t) => - VertexSet.fold - (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst p, z, w) s') (snd p) a)) in *. -rewrite MEdgeFacts.fold_left_assoc. -set (s := fold_left f l EdgeSet.empty) in *. -unfold f. -destruct a. simpl. -rewrite VertexSet.fold_1. -generalize VertexSet.elements_1. -intro H2. generalize (H2 t1 y). clear H2. intro HHH. -induction (VertexSet.elements t1). simpl. -assert (VertexMap.MapsTo x t0 m). -apply VertexMap.find_2. assumption. -generalize (HH H0). intro H2. -inversion H2; subst. -inversion H4. simpl in H3. simpl in H5. subst. -unfold adj_set in H. rewrite H1 in H. -generalize (HHH H). intro. inversion H5. -apply IHl. intro. auto. - -rewrite MEdgeFacts.fold_left_assoc. -generalize (VertexMap.find_2 H1). intro. -generalize (HH H0). clear HH H0. intro HH. -inversion HH; subst. -inversion H2; simpl in *; subst. clear H2 HH. -destruct (Vertex.eq_dec y a). -apply EdgeSet.add_1. -Eq_eq. -apply EdgeSet.add_2. -apply IHl0. -intro H2. generalize (HHH H2). clear HHH H2. intro H2. -inversion H2. subst. -elim (n H4). -assumption. -apply EdgeSet.add_2. -assert (forall l', EdgeSet.In (x,y,w) s -> - EdgeSet.In (x, y, w) - (fold_left - (fun (a0 : EdgeSet.t) (e : VertexSet.elt) => EdgeSet.add (k, e, w) a0) - l' s)). -clear H HH H1 H2 IHl IHl0 HHH Hsym. -intros. -induction l'. simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -destruct (Edge.eq_dec (k,a0,w) (x,y,w)). -apply EdgeSet.add_1. auto. -apply EdgeSet.add_2. apply IHl'. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -apply H0. -apply IHl. -auto. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -unfold f. -intros. set (g := (fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst z, z0, w) s')). -set (g' := fun (z0 : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst y0, z0, w) s'). -apply fold_assoc. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. -unfold g. unfold g'. -intros. apply RegRegProps.add_add. -intros. -unfold f. -rewrite VertexSet.fold_1. -rewrite VertexSet.fold_1. -apply MEdgeFacts.fold_left_compat_set. -assumption. -intros. apply RegRegProps.Dec.F.add_m. apply eq_refl. assumption. - -unfold adj_set in H. rewrite H1 in H. elim (VertexSet.empty_1 H). -Qed. - -Require Import FMapFacts. - -Module InterfFacts := FMapFacts.Facts VertexMap. - -Lemma imap_remove_1 : forall x y m r, -~Vertex.eq r x -> -~Vertex.eq r y -> -VertexSet.In x (adj_set y m) -> -VertexSet.In x (adj_set y (imap_remove m r)). - -Proof. -intros. -unfold imap_remove. -unfold adj_set. -cut (VertexSet.In x match - (VertexMap.find (elt:=VertexSet.t) y - (VertexMap.remove (elt:=VertexSet.t) r - (VertexSet.fold - (fun y0 : VertexSet.elt => - VertexMap.add y0 (VertexSet.remove r (adj_set y0 m))) - (adj_set r m) m))) with - | Some x0 => x0 - | None => VertexSet.empty -end); auto. -rewrite MapFacts.remove_neq_o. - -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set r m)); intros. -simpl. assumption. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. -unfold EqualSetMap in H2. -generalize (H2 y). intro H3. -case_eq (VertexMap.find y (fold_left f (a :: l) m)); intros. -rewrite H4 in H3. inversion H3. subst. -unfold f in H6. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H6. inversion H6. -rewrite H7. rewrite H8. apply VertexSet.remove_2. -auto. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). assumption. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H6. -rewrite H7. unfold f in IHl. -case_eq (VertexMap.find (elt:=VertexSet.t) y - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a) l m)); intros; -rewrite H5 in *. -inversion H6. subst. -assumption. -inversion H6. -auto. -rewrite H4 in H3. inversion H3. subst. -unfold f in H5. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H5. -inversion H5. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H5. -unfold f in IHl. -case_eq (VertexMap.find (elt:=VertexSet.t) y - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a) l m)); intros; -rewrite H6 in *. -inversion H5. -assumption. -auto. -auto. -Qed. - -Lemma imap_remove_2 : forall x y m r, -Vertex.eq r x \/ Vertex.eq r y -> -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -~VertexSet.In x (adj_set y (imap_remove m r)). - -Proof. -intros x y m r H HH. -unfold imap_remove. -unfold adj_set. -cut (~VertexSet.In x match - (VertexMap.find (elt:=VertexSet.t) y - (VertexMap.remove (elt:=VertexSet.t) r - (VertexSet.fold - (fun y0 : VertexSet.elt => - VertexMap.add y0 (VertexSet.remove r (adj_set y0 m))) - (adj_set r m) m))) with - | Some x0 => x0 - | None => VertexSet.empty -end); auto. -destruct (Vertex.eq_dec r y). -rewrite MapFacts.remove_eq_o. -apply VertexSet.empty_1. -assumption. -rewrite MapFacts.remove_neq_o. -destruct H. - -generalize VertexSet.elements_1. intro HHH. -generalize (HHH (adj_set r m) y). clear HHH. intro HHH. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set r m)); intros. -simpl. intro H0. -generalize (HH _ _ H0). intro H1. -assert (InA Vertex.eq y nil). -apply HHH. -unfold adj_set. rewrite (MapFacts.find_o _ H). assumption. -inversion H2. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -unfold EqualSetMap in H0. -generalize (H0 y). intro H1. -case_eq (VertexMap.find y (fold_left f (a :: l) m)); intros. -rewrite H2 in H1. inversion H1. subst. -unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. inversion H4. -rewrite H5. rewrite H6. apply VertexSet.remove_1. auto. apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H4. -rewrite H5. unfold f in IHl. -case_eq (VertexMap.find (elt:=VertexSet.t) y - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a) l m)); intros; -rewrite H3 in *. -inversion H4. subst. -apply IHl. -intros; intuition. -inversion H7; subst. -elim (n0 H9). assumption. -inversion H4. -auto. -apply VertexSet.empty_1. -elim (n H). -auto. -Qed. - -Lemma imap_remove_3 : forall x y m r, -VertexSet.In x (adj_set y (imap_remove m r)) -> -VertexSet.In x (adj_set y m). - -Proof. -intros. -assert (VertexSet.In x match - (VertexMap.find (elt:=VertexSet.t) y - (VertexMap.remove (elt:=VertexSet.t) r - (VertexSet.fold - (fun y0 : VertexSet.elt => - VertexMap.add y0 (VertexSet.remove r (adj_set y0 m))) - (adj_set r m) m))) with - | Some x0 => x0 - | None => VertexSet.empty -end) by auto. generalize H0. clear H H0. intro H. -destruct (Vertex.eq_dec y r). -rewrite MapFacts.remove_eq_o in H. -elim (VertexSet.empty_1 H). apply Regs.eq_sym. auto. -rewrite MapFacts.remove_neq_o in H. -unfold adj_set. -rewrite VertexSet.fold_1 in H. - -generalize VertexSet.elements_2. intro. -generalize (H0 (adj_set y m) x). clear H0. intro HH. - -induction (VertexSet.elements (adj_set r m)). simpl in H. assumption. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -unfold EqualSetMap in H0. -generalize (H0 y). -case_eq (VertexMap.find y (fold_left f (a :: l) m)); intros. -rewrite H1 in *. inversion H2. subst. -unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -apply HH. -apply VertexSet.elements_1. -unfold adj_set. rewrite (MapFacts.find_o _ e). fold (adj_set a m). -apply VertexSet.remove_3 with (x:=r). -inversion H4. subst. rewrite <-H5. assumption. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H4. -fold f in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. assumption. -auto. - -inversion H2. subst. rewrite <-H3 in *. rewrite H1 in *. -elim (VertexSet.empty_1 H). -auto. -Qed. - -Lemma imap_remove_4 : forall x m r, -VertexMap.In x (imap_remove m r) -> -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -VertexMap.In x m. - -Proof. -intros x m r H Hsym. -unfold imap_remove in H. -destruct (Vertex.eq_dec x r). -elim (VertexMap.remove_1 (Vertex.eq_sym e) H). -apply (proj2 (MapFacts.in_find_iff _ _)). -generalize (proj1 (MapFacts.in_find_iff _ _) H). clear H. intro H. -rewrite MapFacts.remove_neq_o in H. -rewrite VertexSet.fold_1 in H. intro H0. elim H. clear H. - -generalize VertexSet.elements_2. intro. -generalize (H (adj_set r m) x). clear H. intro HH. - -induction (VertexSet.elements (adj_set r m)). simpl. -assumption. - -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. -intros. -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. auto. - -generalize (H x). clear H. intro H. -case_eq (VertexMap.find x (fold_left f (a :: l) m)); intros. -rewrite H1 in H. inversion H. subst. -unfold f in H3. -destruct (Vertex.eq_dec x a). -assert (VertexSet.In r (adj_set x m)). -apply Hsym. apply HH. left. auto. -unfold adj_set in H2. rewrite H0 in H2. elim (VertexSet.empty_1 H2). -rewrite MapFacts.add_neq_o in H3. -fold f in H3. -assert (VertexMap.find x (fold_left f l m) = None). -apply IHl. -intro. apply HH. auto. -rewrite H2 in H3. inversion H3. -auto. -reflexivity. -auto. -Qed. - -Lemma imap_remove_5 : forall r x m, -VertexMap.In x m -> -~Vertex.eq x r -> -VertexMap.In x (imap_remove m r). - -Proof. -intros. -unfold imap_remove. -rewrite MapFacts.in_find_iff. -rewrite MapFacts.remove_neq_o. -rewrite VertexSet.fold_1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e m)) a)). -induction (VertexSet.elements (adj_set r m)). simpl. -rewrite MapFacts.in_find_iff in H. assumption. - -cut (VertexMap.find x (f (fold_left f l m) a) <> None). -intro H1. -cut (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). intro H2. -generalize (H2 x). clear H2. intro H2. inversion H2. -simpl. rewrite <-H4 in *. rewrite <-H5 in *. assumption. -simpl. rewrite <-H3 in *. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.remove_m. apply Vertex.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; auto. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold f. -unfold EqualSetMap. -intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Vertex.eq_sym. assumption. -apply Vertex.eq_sym. assumption. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -set (tmp := fold_left f l m) in *. -unfold f. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o. congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -assumption. -auto. -auto. -Qed. - -Lemma extremities_in_remove_vertex_imap v g : -forall x, -VertexMap.In x (imap_remove (imap g) v) <-> -VertexSet.In x (VertexSet.remove v (V g)). - -Proof. -split; intros. -apply VertexSet.remove_2. -unfold imap_remove in H. -intro Helim; apply (VertexMap.remove_1 Helim H). -apply (extremities_imap g x); apply imap_remove_4 with (r:=v); try assumption. -exact (sym_imap g). - -generalize (proj2 (extremities_imap g x)). intro. -generalize (H0 (VertexSet.remove_3 H)). clear H0. intro H0. -apply imap_remove_5. assumption. -intro H1. elim (VertexSet.remove_1 (Vertex.eq_sym H1) H). -Qed. - -Lemma extremities_in_remove_vertex_pmap v g : -forall x, -VertexMap.In x (imap_remove (pmap g) v) <-> -VertexSet.In x (VertexSet.remove v (V g)). - -Proof. -split; intros. -apply VertexSet.remove_2. -unfold imap_remove in H. -intro Helim; apply (VertexMap.remove_1 Helim H). -apply (extremities_pmap g x); apply imap_remove_4 with (r:=v); try assumption. -exact (sym_pmap g). - -generalize (proj2 (extremities_pmap g x)). intro. -generalize (H0 (VertexSet.remove_3 H)). clear H0. intro H0. -apply imap_remove_5. assumption. -intro H1. elim (VertexSet.remove_1 (Vertex.eq_sym H1) H). -Qed. - -Lemma simple_graph_remove_vertex_map v g : -forall x y, -VertexSet.In x (adj_set y (imap_remove (imap g) v)) /\ -VertexSet.In x (adj_set y (imap_remove (pmap g) v)) -> -False. - -Proof. -intros. -apply (simple_graph g x y). -destruct H. -generalize (imap_remove_3 _ _ _ _ H). -generalize (imap_remove_3 _ _ _ _ H0). -auto. -Qed. - -Lemma not_eq_extremities_remove_vertex_map v g : forall x y, -VertexSet.In x (adj_set y (imap_remove (imap g) v)) \/ -VertexSet.In x (adj_set y (imap_remove (pmap g) v)) -> -~Vertex.eq x y. - -Proof. -intros. -apply (not_eq_extremities g). -destruct H;[left|right]; apply (imap_remove_3 _ _ _ _ H). -Qed. - -Lemma sym_imap_remove_vertex v g : -forall (x : VertexSet.elt) (y : VertexMap.key), -VertexSet.In x (adj_set y (imap_remove (imap g) v)) -> -VertexSet.In y (adj_set x (imap_remove (imap g) v)). - -Proof. -intros. -generalize (imap_remove_2 x y (imap g) v). intro H0. -apply imap_remove_1. -intro H1. elim (H0 (or_intror _ H1) (sym_imap g) H). -intro H1. elim (H0 (or_introl _ H1) (sym_imap g) H). -generalize (imap_remove_3 _ _ _ _ H). intro H1. -apply (sym_imap g). assumption. -Qed. - -Lemma sym_pmap_remove_vertex v g : -forall (x : VertexSet.elt) (y : VertexMap.key), -VertexSet.In x (adj_set y (imap_remove (pmap g) v)) -> -VertexSet.In y (adj_set x (imap_remove (pmap g) v)). - -Proof. -intros. -generalize (imap_remove_2 x y (pmap g) v). intro H0. -apply imap_remove_1. -intro H1. elim (H0 (or_intror _ H1) (sym_pmap g) H). -intro H1. elim (H0 (or_introl _ H1) (sym_pmap g) H). -generalize (imap_remove_3 _ _ _ _ H). intro H1. -apply (sym_pmap g). assumption. -Qed. - -Definition remove_vertex v g := -Make_Graph (VertexSet.remove v (V g)) - (imap_remove (imap g) v) - (imap_remove (pmap g) v) - (extremities_in_remove_vertex_imap v g) - (extremities_in_remove_vertex_pmap v g) - (simple_graph_remove_vertex_map v g) - (sym_imap_remove_vertex v g) - (sym_pmap_remove_vertex v g) - (not_eq_extremities_remove_vertex_map v g). - -Definition map_merge e map := -let adj_snd := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) map) in -let adj_fst := VertexSet.remove (snd_ext e) (adj_set (fst_ext e) map) in -let new_fst_adj := VertexSet.union adj_fst adj_snd in -(* -let new_fst_adj_ := VertexSet.union (adj_set (fst_ext e) map) (adj_set (snd_ext e) map) in -let new_fst_adj := VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) new_fst_adj_) in -let m := VertexMap.add (fst_ext e) new_fst_adj map in -*) -let redirect_m := - VertexSet.fold - (fun y m' => - VertexMap.add y - (VertexSet.add (fst_ext e) (VertexSet.remove (snd_ext e) (adj_set y map))) m') - adj_snd -(* - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) -*) - map in -VertexMap.remove (snd_ext e) -(VertexMap.add (fst_ext e) new_fst_adj redirect_m). - -Definition imap_merge e g := map_merge e (imap g). - -Definition resolve_conflicts y pm padj iadj := -let m' := VertexSet.fold - (fun x m => VertexMap.add x (VertexSet.remove y (adj_set x pm)) m) - (VertexSet.inter padj iadj) - pm -in VertexMap.add y (VertexSet.diff padj iadj) m'. - -Definition pmap_merge e g im := -let pm := map_merge e (pmap g) in -resolve_conflicts (fst_ext e) pm (adj_set (fst_ext e) pm) (adj_set (fst_ext e) im). - -Definition In_graph_edge e g := -EdgeSet.In e (AE g) \/ EdgeSet.In e (IE g). - -Lemma In_graph_edge_dec : forall e g, -{In_graph_edge e g}+{~In_graph_edge e g}. - -Proof. -intros e g. -destruct (RegRegProps.In_dec e (AE g)). -left. left. assumption. -destruct (RegRegProps.In_dec e (IE g)). -left. right. assumption. -right. intro H. destruct H;[elim n|elim n0];assumption. -Qed. - -Lemma aff_edge_dec : forall e, -{aff_edge e}+{~aff_edge e}. - -Proof. -intro e. -case_eq (get_weight e). -intros n H. left. unfold aff_edge. exists n. auto. -intro H. right. intro H0. unfold aff_edge in H0. -destruct H0 as [w H0]. rewrite H0 in H. inversion H. -Qed. - -Definition In_graph (v : Vertex.t) g := VertexSet.In v (V g). - -Add Morphism get_weight : get_weight_m. - -Proof. -intros x y H. -rewrite (weight_ordered_weight x);rewrite (weight_ordered_weight y). -unfold get_weight;unfold E.eq in H. -destruct H as [_ H];inversion H;[|rewrite H2];reflexivity. -Qed. - -Lemma E_weights_aux : forall e map w s, -EdgeSet.In e -(VertexMap.fold - (fun y imapy s => VertexSet.fold - (fun z s' => EdgeSet.add (y,z,w) s') - imapy - s) - map - s) -> -EdgeSet.In e s \/ get_weight e = w. - -Proof. -intros. -rewrite VertexMap.fold_1 in H. -generalize H. clear H. generalize (VertexMap.elements map) s. -induction l. -simpl. auto. -intros. simpl in H. -set (s' := VertexSet.fold - (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst a, z, w) s') (snd a) s0) in H. -generalize (IHl s' H). intro H0. -destruct H0. -unfold s' in H0. - -assert (EdgeSet.In e (VertexSet.fold (fun (z : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst a, z, w) s') (snd a) s0) -> EdgeSet.In e s0 \/ get_weight e = w). -clear H IHl. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (snd a)). -simpl. auto. -rewrite MEdgeFacts.fold_left_assoc. -intro H1. destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H1). -fold (eq (fst a, a0, w) e) in H. -right. rewrite <-H. simpl. reflexivity. -apply IHl0. assumption. - -intros. apply RegRegProps.add_add. -intros. apply RegRegProps.Dec.F.add_m. -apply eq_refl. -assumption. -apply H1. assumption. -right. assumption. -Qed. - -Lemma E_weights : forall e m w, -EdgeSet.In e (edgemap_to_edgeset m w) -> get_weight e = w. - -Proof. -intros. -generalize (E_weights_aux e m w EdgeSet.empty). intro H0. -destruct H0. -assumption. -elim (EdgeSet.empty_1 H0). -assumption. -Qed. - -Lemma IE_weights : forall g e, -EdgeSet.In e (IE g) -> get_weight e = None. - -Proof. -unfold IE. intros. eapply E_weights. eassumption. -Qed. - -Lemma AE_weights : forall g e, -EdgeSet.In e (AE g) -> get_weight e = Some N0. - -Proof. -unfold AE. intros. eapply E_weights. eassumption. -Qed. - -(* extremities of edges are in the graph *) -Lemma In_graph_edge_in_ext : forall e g, -In_graph_edge e g -> In_graph (fst_ext e) g /\ In_graph (snd_ext e) g. - -Proof. -intros. -split. destruct H. -apply (proj1 (extremities_pmap g (fst_ext e))). - -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (fst_ext e) (pmap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). - -apply (proj1 (extremities_imap g (fst_ext e))). -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (fst_ext e) (imap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). - -destruct H. -apply (proj1 (extremities_pmap g (snd_ext e))). -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e) in H. rewrite edge_comm in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (snd_ext e) (pmap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). - -apply (proj1 (extremities_imap g (snd_ext e))). -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -rewrite (edge_eq e) in H. rewrite edge_comm in H. -simpl in Hw. rewrite Hw in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H). intro H0. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H0. -case_eq (VertexMap.find (snd_ext e) (imap g)); intros; rewrite H1 in H0. -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H0). -Qed. - -Lemma not_eq_extremities_map_merge : forall x y e m, -(forall a b, VertexSet.In a (adj_set b m) -> ~Vertex.eq a b) -> -VertexSet.In x (adj_set y (map_merge e m)) -> -~Vertex.eq x y. - -Proof. -intros x y e m Hsimp H. -unfold map_merge in H. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)))) in *. -set (s' := (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m))) in *. -intro. -unfold adj_set in H. rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -rewrite H0 in H1. rewrite e0 in H1. -elim (Hsimp _ _ H1). auto. -unfold s' in H1. -rewrite H0 in H1. rewrite e0 in H1. elim (VertexSet.remove_1 (Vertex.eq_refl _) H1). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements s'). simpl in H. -unfold adj_set in H. -fold (adj_set y m) in H. -elim (Hsimp _ _ H). assumption. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 y). clear H1. intro H1. simpl in H. inversion H1; clear H1. -unfold adj_set in H. rewrite <-H3 in H. elim (VertexSet.empty_1 H). - -rewrite <-H2 in H. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. -rewrite H4 in H. clear H4. -rewrite H0 in H. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (n (Vertex.eq_sym H1)). -generalize (VertexSet.remove_3 H1). intro. -elim (Hsimp _ _ H3). auto. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl. -apply IHl. rewrite <-H4. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. -auto. - -auto. - -intro. rewrite MapFacts.remove_eq_o in H. elim (VertexSet.empty_1 H). auto. -Qed. - -Lemma resolve_conflicts_map_0 : forall x y e g, -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) -> -VertexSet.In x (adj_set y (map_merge e (pmap g))). - -Proof. -intros. -unfold resolve_conflicts in H. -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x (VertexSet.remove (fst_ext e) (adj_set x (map_merge e (pmap g)))) m)) in *. -set (s := (VertexSet.diff (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (inter := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -unfold adj_set in H. -destruct (Vertex.eq_dec y (fst_ext e)). rewrite MapFacts.add_eq_o in H. -unfold s in H. -generalize (VertexSet.diff_1 H). intro H0. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -apply Regs.eq_sym. auto. - -rewrite MapFacts.add_neq_o in H. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -set (m := map_merge e (pmap g)) in *. -induction (VertexSet.elements inter). simpl in H. assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H0 y). clear H0. simpl in H. intro H0. inversion H0; clear H0. -rewrite <-H2 in H. elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H2. -rewrite <-H1 in H. rewrite H3 in H. clear H3. -inversion H2; subst; clear H2. -unfold adj_set. rewrite (MapFacts.find_o _ e0). -apply (VertexSet.remove_3 H). -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H2. -apply IHl. rewrite <-H2. rewrite <-H1 in H. rewrite <-H3. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. -auto. -auto. -Qed. - -Lemma pmap_merge_sub : forall x y e g, -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -VertexSet.In x (adj_set y (map_merge e (pmap g))). - -Proof. -exact (fun x y e g H => resolve_conflicts_map_0 _ _ _ _ H). -Qed. - -Lemma pmap_merge_domain_1 : forall x e g, -In_graph_edge e g -> -VertexMap.In x (map_merge e (pmap g)) -> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g)). - -Proof. -intros x e g HH H. -unfold map_merge in H. -set (m := pmap g) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -set (adj := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) m)) s)) in *. -set (f := (fun (y : VertexSet.elt) (m'0 : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m'0)) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -rewrite MapFacts.in_find_iff in H. -assert (~Vertex.eq x (snd_ext e)). -intro. rewrite MapFacts.remove_eq_o in H. congruence. apply Regs.eq_sym. auto. -apply VertexSet.remove_2. auto. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite e0. apply (proj1 (In_graph_edge_in_ext _ _ HH)). - -cut (forall z, VertexSet.In z s -> VertexSet.In z (V g)). intro HHH. -generalize VertexSet.elements_2. intro H1. -generalize (H1 s x). clear H1. intro Helt. - -induction (VertexSet.elements s). simpl in H. -apply (proj1 (extremities_pmap g x)). -rewrite MapFacts.in_find_iff. -rewrite MapFacts.add_neq_o in H. -assumption. -auto. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl in H. inversion H1. -rewrite MapFacts.add_neq_o in H. -rewrite <-H3 in H. congruence. auto. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -apply HHH. apply Helt. left. auto. -rewrite MapFacts.add_neq_o in H3. -apply IHl. -rewrite MapFacts.add_neq_o. -congruence. -auto. -intro. apply Helt. right. auto. auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. -auto. -auto. - -intros. -unfold s in H1. -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -apply (proj1 (extremities_pmap g z)). -rewrite MapFacts.in_find_iff. -generalize (sym_pmap g _ _ H1). clear H1. intro H1. -unfold adj_set in H1. -destruct (VertexMap.find z (pmap g)). -congruence. -elim (VertexSet.empty_1 H1). -auto. -Qed. - -Lemma pmap_merge_domain_2 : forall x e g, -In_graph_edge e g -> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g)) -> -VertexMap.In x (map_merge e (pmap g)). - -Proof. -intros x e g HH H. -unfold map_merge. -set (m := pmap g) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -set (adj := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) m)) s)) in *. -set (f := (fun (y : VertexSet.elt) (m'0 : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m'0)) in *. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -rewrite MapFacts.in_find_iff. -assert (~Vertex.eq x (snd_ext e)). -intro. elim (VertexSet.remove_1 (Vertex.eq_sym H0) H). -generalize (VertexSet.remove_3 H). clear H. intro H. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. congruence. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. - -induction (VertexSet.elements s). simpl. -rewrite <-MapFacts.in_find_iff. -apply (proj2 (extremities_pmap g x)). assumption. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl. inversion H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. congruence. apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o in H4. congruence. auto. -congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. unfold f'. unfold f. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. -auto. -auto. - -auto. -auto. -Qed. - -(* there is no loop in the graph *) -Lemma In_graph_edge_diff_ext : forall e g, -In_graph_edge e g -> ~Vertex.eq (snd_ext e) (fst_ext e). - -Proof. -intros. -apply (not_eq_extremities g). -destruct H;[right|left]. - -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). - -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -rewrite (edge_eq e) in H. -simpl in Hw. rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_imap g)) H). -Qed. - -Lemma extremities_in_merge_map e g : -In_graph_edge e g -> -(forall x, -VertexMap.In x (map_merge e (pmap g)) <-> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g))). - -Proof. -intros e g HH x; split; intro H0. -generalize H0. intro H. -unfold map_merge in H0. -apply VertexSet.remove_2. -intro H1. -elim (VertexMap.remove_1 H1 H0). -generalize (proj1 (MapFacts.remove_in_iff _ _ _) H0). clear H0. -intro H0. destruct H0. -generalize (proj1 (MapFacts.add_in_iff _ _ _ _) H1). clear H1. -intro H1. -destruct H1. -rewrite <-H1. -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -apply (proj1 (extremities_pmap g x)). -rewrite VertexSet.fold_1 in H1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (pmap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))) in *. -cut (forall z, In z (VertexSet.elements s) -> - VertexMap.In z (pmap g)). -intros HHH. -induction (VertexSet.elements s). -simpl in H1. assumption. -set (m := pmap g) in *. -cut (VertexMap.In x (f (fold_left f l m) a)). -clear H1. intro H1. -set (tmp := fold_left f l m) in *. -unfold f in H1. -destruct (Vertex.eq_dec x a). -rewrite e0. apply (HHH a). left. auto. -rewrite MapFacts.in_find_iff in H1. -rewrite MapFacts.add_neq_o in H1. -apply IHl. -rewrite MapFacts.in_find_iff. -assumption. -intros. apply HHH. right. assumption. -auto. - -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_refl. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_refl. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_refl. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_refl. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. -auto. -rewrite MapFacts.in_find_iff. -generalize (H2 x). intro. -inversion H3. -rewrite MapFacts.in_find_iff in H1. -simpl in H1. rewrite <-H5 in H1. assumption. -congruence. - -intros. -exists (adj_set z (pmap g)). apply VertexMap.elements_2. -apply VertexMap.elements_1. -unfold adj_set. -case_eq (VertexMap.find z (pmap g)); intros. -apply VertexMap.find_2. assumption. - -assert (VertexSet.In z s). -apply VertexSet.elements_2. -apply In_InA. apply Regs.eq_refl. assumption. -unfold s in H4. -generalize (VertexSet.remove_3 H4). clear H4. intro H4. -generalize (sym_pmap g _ _ H4). clear H4. intro H4. -unfold adj_set in H4. -rewrite H3 in H4. elim (VertexSet.empty_1 H4). - -unfold map_merge. -rewrite MapFacts.in_find_iff. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. -congruence. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (pmap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))) in *. -induction (VertexSet.elements s). simpl. -generalize (proj2 (extremities_pmap g x) (VertexSet.remove_3 H0)). -rewrite MapFacts.in_find_iff. auto. - -set (m := pmap g) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_sym. auto. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_sym. auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -apply Regs.eq_sym. auto. -apply RegFacts.Props.Dec.F.remove_m. -apply Regs.eq_refl. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -apply Regs.eq_sym. auto. -apply Regs.eq_sym. auto. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. -auto. - -generalize (H x). clear H. intro H. -inversion H. -simpl. -set (tmp := fold_left f l m) in *. -unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -congruence. -intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite H3 in IHl. congruence. -auto. -simpl. rewrite <-H1. congruence. -auto. -intro. elim (VertexSet.remove_1 H H0). -Qed. - -Lemma extremities_in_merge_imap e g : -In_graph_edge e g -> -(forall x, -VertexMap.In x (imap_merge e g) <-> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g))). - -Proof. -intros e g HH x; split; intro H0. -generalize H0. intro H. -unfold imap_merge in H0. -unfold map_merge in H0. -apply VertexSet.remove_2. -intro H1. -elim (VertexMap.remove_1 H1 H0). -generalize (proj1 (MapFacts.remove_in_iff _ _ _) H0). clear H0. -intro H0. destruct H0. -generalize (proj1 (MapFacts.add_in_iff _ _ _ _) H1). clear H1. -intro H1. -destruct H1. -rewrite <-H1. -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -apply (proj1 (extremities_imap g x)). -rewrite VertexSet.fold_1 in H1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (imap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))) in *. -cut (forall z, In z (VertexSet.elements s) -> - VertexMap.In z (imap g)). -intros HHH. -induction (VertexSet.elements s). -simpl in H1. assumption. -set (m := imap g) in *. -cut (VertexMap.In x (f (fold_left f l m) a)). -clear H1. intro H1. -set (tmp := fold_left f l m) in *. -unfold f in H1. -destruct (Vertex.eq_dec x a). -rewrite e0. apply (HHH a). left. auto. -rewrite MapFacts.in_find_iff in H1. -rewrite MapFacts.add_neq_o in H1. -apply IHl. -rewrite MapFacts.in_find_iff. -assumption. -intros. apply HHH. right. assumption. -auto. - -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. -auto. -rewrite MapFacts.in_find_iff. -generalize (H2 x). intro. -inversion H3. -rewrite MapFacts.in_find_iff in H1. -simpl in H1. rewrite <-H5 in H1. assumption. -congruence. - -intros. -exists (adj_set z (imap g)). apply VertexMap.elements_2. -apply VertexMap.elements_1. -unfold adj_set. -case_eq (VertexMap.find z (imap g)); intros. -apply VertexMap.find_2. assumption. - -assert (VertexSet.In z s). -apply VertexSet.elements_2. -apply In_InA. intuition. auto. -unfold s in H4. -generalize (VertexSet.remove_3 H4). clear H4. intro H4. -generalize (sym_imap g _ _ H4). clear H4. intro H4. -unfold adj_set in H4. -rewrite H3 in H4. elim (VertexSet.empty_1 H4). - -unfold imap_merge. -unfold map_merge. -rewrite MapFacts.in_find_iff. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. -congruence. -intuition. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f := (fun (a : VertexMap.t VertexSet.t) (e0 : VertexSet.elt) => - VertexMap.add e0 - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set e0 (imap g)))) a)) in *. -set (s := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))) in *. -induction (VertexSet.elements s). simpl. -generalize (proj2 (extremities_imap g x) (VertexSet.remove_3 H0)). -rewrite MapFacts.in_find_iff. auto. - -set (m := imap g) in *. -assert (EqualSetMap (fold_left f (a :: l) m) (f (fold_left f l m) a)). -apply fold_left_assoc_map. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ e1). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply RegFacts.Props.Dec.F.add_m. -intuition. -apply RegFacts.Props.Dec.F.remove_m. -intuition. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. -auto. - -generalize (H x). clear H. intro H. -inversion H. -simpl. -set (tmp := fold_left f l m) in *. -unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -congruence. -intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite H3 in IHl. congruence. -intuition. -simpl. rewrite <-H1. congruence. -auto. -intro. elim (VertexSet.remove_1 H H0). -Qed. - -Lemma sym_map_merge_map : forall e g m x y, -aff_edge e -> -In_graph_edge e g -> -(forall a b, VertexSet.In a (adj_set b m) -> - VertexSet.In b (adj_set a m)) -> -(forall a b, VertexSet.In a (adj_set b (map_merge e m)) -> - ~Vertex.eq a b) -> -~Vertex.eq x (snd_ext e) -> -~Vertex.eq y (snd_ext e) -> -VertexSet.In x (adj_set y (map_merge e m)) -> -VertexSet.In y (adj_set x (map_merge e m)). - -Proof. -intros e g m x y Haff Hin Hsym Hnoteq Hsndx Hsndy H. -assert (~Vertex.eq x y) as Hdiff. -apply Hnoteq. assumption. -unfold map_merge in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) m)))) in *. -set (s' := adj_set (snd_ext e) m) in *. -cut (forall z, VertexSet.In z s' -> ~Vertex.eq z (fst_ext e) -> VertexSet.In z s). intro Himp. -unfold adj_set. rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -rewrite e1. rewrite <-e0. assumption. -intuition. - -rewrite MapFacts.add_neq_o in H. -case_eq (VertexMap.find y (VertexSet.fold f (VertexSet.remove (fst_ext e) s') m)); intros. -rewrite H0 in H. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. - -assert (eq_set_option (VertexMap.find (elt:=VertexSet.t) y - (fold_left f' (VertexSet.elements (VertexSet.remove (fst_ext e) s')) - m)) (Some t0)). rewrite H0. constructor. apply VertexSet.eq_refl. -generalize H1. clear H0 H1. intro H0. -generalize VertexSet.elements_2. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s')). clear H1. intro HH. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -unfold s. apply VertexSet.union_2. -apply VertexSet.remove_2. -auto. -apply Hsym. rewrite <-e0. -unfold adj_set. inversion H0. subst. rewrite H3. assumption. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 y). clear H1. intro H1. simpl in H0. inversion H1. -inversion H0. congruence. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). -rewrite e1. -apply Himp. -apply VertexSet.remove_3 with (x:= fst_ext e). -apply HH. left. intuition. -assert (VertexSet.In a (VertexSet.remove (fst_ext e) s')). -apply HH. left. intuition. -intro. elim (VertexSet.remove_1 (Vertex.eq_sym H6) H5). -rewrite MapFacts.add_neq_o in H3. clear H1. -apply IHl. -inversion H0. subst. -rewrite <-H1 in H2. clear H1. inversion H2. subst. clear H2. -rewrite <-H3. constructor. -rewrite <-H4. assumption. - -intros. apply HH. right. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -rewrite H0 in H. -elim (VertexSet.empty_1 H). -auto. -auto. -intuition. - -rewrite MapFacts.add_neq_o. -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. - -rewrite VertexSet.fold_1. -generalize VertexSet.elements_1. intro H0. -generalize (H0 (VertexSet.remove (fst_ext e) s') x). clear H0. intro H0. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl. -unfold s in H. -destruct (VertexSet.union_1 H). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -generalize (Hsym _ _ H1). clear H1. intro H1. -rewrite <-e0 in H1. assumption. -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -assert (VertexSet.In x (VertexSet.remove (fst_ext e) s')) by - (apply VertexSet.remove_2; auto). -generalize (H0 H2). intro H3. inversion H3. -set (f' := fun a e => f e a) in *. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl. inversion H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. congruence. -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. intro. -generalize (H0 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H6). -assumption. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. clear H1. -rewrite H4. apply VertexSet.add_1. intuition. intuition. -rewrite MapFacts.add_neq_o in H3. rewrite <-H3 in IHl. -rewrite H4. apply IHl. intro. -generalize (H0 H5). clear H5. intro H5. -inversion H5; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -intuition. - -rewrite MapFacts.add_neq_o in H. -rewrite VertexSet.fold_1 in H. rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl in H. simpl. -apply (Hsym _ _ H). - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H0 x). generalize (H0 y). clear H0. intros H0 HH0. simpl. simpl in H. inversion H0. -rewrite <-H2 in H. elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H2. -inversion HH0. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec x a). -elim Hdiff. apply (Vertex.eq_trans e1 (Vertex.eq_sym e0)). -rewrite MapFacts.add_neq_o in H6. - -cut (VertexMap.In x tmp). -intro H7. rewrite MapFacts.in_find_iff in H7. congruence. -cut (VertexMap.In x m -> VertexMap.In x tmp). -intro H7. apply H7. clear H7. -rewrite MapFacts.in_find_iff. intro. - -rewrite <-H1 in *. rewrite H3 in H. clear H0 HH0 H1 H5 H6 IHl. -inversion H2. subst. clear H2. clear H3. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). elim n. auto. -generalize (VertexSet.remove_3 H0). clear H H0. intro H. -generalize (Hsym _ _ H). clear H. intro H. -unfold adj_set in H. rewrite H4 in H. elim (VertexSet.empty_1 H). - -clear IHl H H0 HH0 H1 H2 H3 H5 H6. intro. -unfold tmp. -induction l. simpl. assumption. - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H0 x). clear H0. simpl. intro H0. inversion H0. -set (tmp' := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H3. congruence. intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff in IHl. congruence. intuition. -rewrite MapFacts.in_find_iff. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s1); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -auto. -clear H0 HH0. -rewrite <-H1 in H. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec x a). -elim Hdiff. apply Vertex.eq_trans with (y:=a); auto. -rewrite MapFacts.add_neq_o in H5. -rewrite <-H5 in *. -inversion H2. subst. clear H2. -rewrite H3 in H. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). elim n. auto. -generalize (VertexSet.remove_3 H0). clear H H0. intro H. - -case_eq (VertexMap.find y tmp); intros. -rewrite H0 in IHl. -rewrite H6. apply IHl. - -clear IHl H1 H6 H4 H3 H5. -unfold tmp in H0. -unfold adj_set in H. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)) in H. -assert (eq_set_option (VertexMap.find y (fold_left f' l m)) (Some t0)). -rewrite H0. constructor. apply VertexSet.eq_refl. -generalize H1. clear H0 H1. intro H0. -induction l. simpl in H0. inversion H0. subst. clear H0. rewrite <-H1 in H. rewrite <-H3. assumption. -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H1 y). clear H1. simpl in H0. intro H1. inversion H1. clear H1. -set (tmp' := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -rewrite MapFacts.add_neq_o in H4. - -case_eq (VertexMap.find y m); intros. -clear H. - -assert (VertexMap.In y tmp'). -clear H3 H4 IHl H0. -unfold tmp'. induction l. simpl. -rewrite MapFacts.in_find_iff. congruence. - -cut (EqualSetMap (fold_left f' (a1 :: l) m) (f' (fold_left f' l m) a1)). intro. -generalize (H y). clear H. intro H. simpl. inversion H. clear H. -set (tmp'' := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a1). -rewrite MapFacts.add_eq_o in H3. congruence. intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite MapFacts.in_find_iff in IHl. rewrite <-H3 in IHl. congruence. -auto. -rewrite MapFacts.in_find_iff. rewrite <-H0. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a2). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H. auto. auto. - -rewrite MapFacts.in_find_iff in H. congruence. -rewrite H1 in H. elim (VertexSet.empty_1 H). -auto. - -rewrite <-H2 in *. inversion H0. subst. clear H0 H1. - -set (tmp' := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a0). rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. -case_eq (VertexMap.find y m); intros. -rewrite H0 in H. -rewrite <-H7. rewrite H4. -apply VertexSet.add_2. -apply VertexSet.remove_2. -auto. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite H0. assumption. -rewrite H0 in H. elim (VertexSet.empty_1 H). -intuition. -rewrite MapFacts.add_neq_o in H3. - -apply IHl. -rewrite <-H3. constructor. -rewrite <-H4. rewrite <-H7. apply VertexSet.eq_refl. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -unfold adj_set in H. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)) in H. -clear IHl H1 H6 H4 H5 H3. -unfold tmp in H0. -induction l. simpl in H0. -rewrite H0 in H. elim (VertexSet.empty_1 H). - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H1 y). clear H1. intro H1. simpl in H0. inversion H1. -set (tmp' := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -rewrite MapFacts.add_neq_o in H4. -apply IHl. auto. auto. - -congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -auto. -intuition. -rewrite MapFacts.add_neq_o in H2. -rewrite <-H2 in *. rewrite <-H1 in *. - -inversion HH0. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H6. congruence. intuition. -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. -apply IHl. rewrite <-H3. assumption. -auto. - -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H5. -clear H0 HH0. - -inversion H5. subst. clear H5. -rewrite H6. -apply VertexSet.add_2. -apply VertexSet.remove_2. -auto. -apply Hsym. -rewrite <-e0. -rewrite H3 in H. clear IHl H1 H4 H6. -unfold tmp in H2. -assert (eq_set_option (Some s'0) (VertexMap.find y (fold_left f' l m))). -rewrite <-H2. constructor. apply VertexSet.eq_refl. -generalize H0. clear H0 H2. intro H2. -induction l. simpl in H2. -unfold adj_set. inversion H2. subst. rewrite <-H4. assumption. - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H0 y). clear H0. simpl in H2. intro H0. inversion H0. clear H0. -rewrite <-H4 in H2. inversion H2. -set (tmp' := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H4. inversion H4. subst. clear H4. -inversion H2. subst. rewrite <-H6 in H1. inversion H1. subst. -apply VertexSet.remove_3 with (x:=snd_ext e). -apply VertexSet.add_3 with (x:= fst_ext e). -auto. -unfold adj_set. rewrite (MapFacts.find_o _ e1). rewrite <-H5. rewrite <-H7. assumption. -intuition. -rewrite MapFacts.add_neq_o in H4. -apply IHl. -rewrite <-H4. constructor. rewrite <-H1 in H2. inversion H2. subst. -rewrite H8. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. -intuition. -rewrite MapFacts.add_neq_o in H5. -rewrite <-H5 in *. -rewrite H6. apply IHl. -rewrite <-H3. assumption. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -auto. -auto. -auto. -auto. - -clear H. -unfold s. unfold s'. intros z H Hne. -apply VertexSet.union_3. apply VertexSet.remove_2; auto. -Qed. - -Lemma not_eq_extremities_merge_map : forall e g, -aff_edge e -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) \/ -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -~Vertex.eq x y. - -Proof. -intros. destruct H0. -apply (not_eq_extremities_map_merge x y e (imap g)). -intros. apply (not_eq_extremities g). left. assumption. assumption. -apply (not_eq_extremities_map_merge x y e (pmap g)). -intros. apply (not_eq_extremities g). right. assumption. -apply pmap_merge_sub. assumption. -Qed. - -Lemma sym_merge : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -VertexSet.In y (adj_set x (map_merge e (pmap g))). - -Proof. -intros e g Haff Hin x y H. -apply sym_map_merge_map with (g:=g); auto. -apply (sym_pmap g). -intros. apply (not_eq_extremities_map_merge _ _ e (pmap g)). -intros. apply (not_eq_extremities g). right. auto. -assumption. - -assert (~Vertex.eq y (snd_ext e)) as Hy. -intro. -generalize (extremities_in_merge_map e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (map_merge e (pmap g))); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). - -intro. -unfold map_merge in H. - -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y (pmap g)))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) (pmap g))))) in *. -set (s' := adj_set (snd_ext e) (pmap g)) in *. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold s' in H1. - -elim (not_eq_extremities g x (snd_ext e)). -right. assumption. -assumption. -intuition. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s') y). clear H1. intro H1. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -simpl in H. -unfold s' in H1. -assert (InA Vertex.eq y nil). -apply H1. -apply VertexSet.remove_2. auto. -apply (sym_pmap g). rewrite <-H0. assumption. inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) (pmap g)) (f' (fold_left f' l (pmap g)) a)). intro. -generalize (H2 y). clear H2. intro H2. simpl in H. inversion H2; clear H2. -rewrite <-H4 in *. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l (pmap g)) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -rewrite <-H3 in *. clear H3. inversion H4. subst. clear H4. -rewrite H5 in H. clear H5. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (In_graph_edge_diff_ext _ _ Hin). -apply Vertex.eq_trans with (y := x); auto. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H2). -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. rewrite <-H3 in H. assumption. -intro. generalize (H1 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -auto. - -intro. -generalize (extremities_in_merge_map e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (map_merge e (pmap g))); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). -Qed. - -Lemma sym_imap_merge_map : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) -> -VertexSet.In y (adj_set x (imap_merge e g)). - -Proof. -intros e g Haff Hin x y H. -apply sym_map_merge_map with (g:=g); auto. -apply (sym_imap g). -intros. apply (not_eq_extremities_merge_map e g Haff). left. assumption. - -assert (~Vertex.eq y (snd_ext e)) as Hy. -intro. -generalize (extremities_in_merge_imap e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (imap_merge e g)); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). - -intro. -unfold imap_merge in H. -unfold map_merge in H. - -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y (imap g)))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) (imap g))))) in *. -set (s' := adj_set (snd_ext e) (imap g)) in *. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold s' in H1. - -elim (not_eq_extremities g x (snd_ext e)). -left. assumption. -assumption. -intuition. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s') y). clear H1. intro H1. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -simpl in H. -unfold s' in H1. -assert (InA Vertex.eq y nil). -apply H1. -apply VertexSet.remove_2. auto. -apply (sym_imap g). rewrite <-H0. assumption. inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) (imap g)) (f' (fold_left f' l (imap g)) a)). intro. -generalize (H2 y). clear H2. intro H2. simpl in H. inversion H2; clear H2. -rewrite <-H4 in *. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l (imap g)) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -rewrite <-H3 in *. clear H3. inversion H4. subst. clear H4. -rewrite H5 in H. clear H5. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (In_graph_edge_diff_ext _ _ Hin). -apply Vertex.eq_trans with (y := x); auto. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H2). -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. rewrite <-H3 in H. assumption. -intro. generalize (H1 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -auto. - -intro. -generalize (extremities_in_merge_imap e g Hin y). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. -case_eq (VertexMap.find y (imap_merge e g)); intros. -rewrite H2 in H1. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -rewrite <-H1. congruence. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H3). -rewrite H2 in H. elim (VertexSet.empty_1 H). -Qed. - -Lemma pmap_merge_in_merge : forall x e g, -aff_edge e -> -In_graph_edge e g -> -VertexMap.In x (pmap_merge e g (imap_merge e g)) -> -VertexMap.In x (map_merge e (pmap g)). - -Proof. -intros x e g Haff HH H. -unfold pmap_merge in H. -unfold resolve_conflicts in H. -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x - (VertexSet.remove (fst_ext e) - (adj_set x (map_merge e (pmap g)))) m)) in *. -rewrite MapFacts.in_find_iff in *. intro H0. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite (MapFacts.find_o _ e0) in H0. -assert (VertexMap.In (fst_ext e) (map_merge e (pmap g))). -apply pmap_merge_domain_2. assumption. -apply VertexSet.remove_2. apply (In_graph_edge_diff_ext _ _ HH). -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -rewrite MapFacts.in_find_iff in H1. congruence. -rewrite MapFacts.add_neq_o in H. -set (s' := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (m := map_merge e (pmap g)) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -assert (forall z, InA Vertex.eq z (VertexSet.elements s') -> - VertexMap.In z m) as Hin. -intros. -apply pmap_merge_domain_2. assumption. -generalize (VertexSet.elements_2 H1). clear H1. intro H1. -unfold s' in H1. generalize (VertexSet.inter_1 H1). -generalize (VertexSet.inter_2 H1). clear H1. intros. -rewrite <-(extremities_in_merge_imap e g HH). -generalize (sym_imap_merge_map e g Haff HH _ _ H1). intro. -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H3. rewrite H4 in H3. elim (VertexSet.empty_1 H3). -induction (VertexSet.elements s'). simpl in H. congruence. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. simpl in H. intro H1. inversion H1; clear H1. -congruence. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. inversion H3; subst; clear H3. -rewrite <-H2 in H. -rewrite (MapFacts.find_o _ e0) in H0. -assert (VertexMap.In a m). -apply Hin. left. auto. -rewrite MapFacts.in_find_iff in H1. congruence. -intuition. -rewrite MapFacts.add_neq_o in H3. -apply IHl. congruence. -intros. apply Hin. right. auto. -auto. - - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -auto. -Qed. - -Lemma pmap_merge_merge_in : forall x e g, -In_graph_edge e g -> -VertexMap.In x (map_merge e (pmap g)) -> -VertexMap.In x (pmap_merge e g (imap_merge e g)). - -Proof. -intros x e g HH H. -unfold pmap_merge. -unfold resolve_conflicts. -set (f := (fun (x0 : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x0 - (VertexSet.remove (fst_ext e) (adj_set x0 (map_merge e (pmap g)))) - m)) in *. -set (s := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -rewrite MapFacts.in_find_iff. -destruct (Vertex.eq_dec x (fst_ext e)). -rewrite MapFacts.add_eq_o. congruence. -intuition. -rewrite MapFacts.add_neq_o. -intro H0. -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -assert (VertexMap.find x (map_merge e (pmap g)) = None). -induction (VertexSet.elements s). simpl in H0. assumption. -set (m := map_merge e (pmap g)) in *. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 x). clear H1. intro H1. simpl in H0. inversion H1; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). rewrite MapFacts.add_eq_o in H4. congruence. -intuition. -rewrite MapFacts.add_neq_o in H4. -apply IHl. auto. -auto. - -congruence. -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -auto. - -rewrite MapFacts.in_find_iff in H. congruence. -auto. -Qed. - -Lemma extremities_in_merge_pmap e g : -aff_edge e -> -In_graph_edge e g -> -(forall x, -VertexMap.In x (pmap_merge e g (imap_merge e g)) <-> -VertexSet.In x (VertexSet.remove (snd_ext e) (V g))). - -Proof. -intros e g Haff HH x;split; intro H0. -apply pmap_merge_domain_1. assumption. -apply pmap_merge_in_merge. assumption. -assumption. assumption. -apply pmap_merge_merge_in. assumption. -apply pmap_merge_domain_2. assumption. -assumption. -Qed. - -Lemma merge_conflicts_aux_1 : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) -> -~Vertex.eq y (fst_ext e) -> -~Vertex.eq x (fst_ext e) -> -VertexSet.In x (adj_set y (imap g)). - -Proof. -intros. -unfold imap_merge in H1. -unfold map_merge in H1. -set (m := imap g) in *. -set (f := fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) - m') in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) m)))) in *. -set (s' := (adj_set (snd_ext e) m)) in *. -unfold adj_set in H1. -destruct (Vertex.eq_dec y (snd_ext e)). -rewrite MapFacts.remove_eq_o in H1. elim (VertexSet.empty_1 H1). -intuition. -rewrite MapFacts.remove_neq_o in H1. rewrite MapFacts.add_neq_o in H1. -rewrite VertexSet.fold_1 in H1. set (f' := fun a e => f e a) in *. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl in H1. -assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H4 y). clear H4. intro H4. simpl in H1. inversion H4; subst; clear H4. -rewrite <-H6 in H1. elim (VertexSet.empty_1 H1). -set (tmp := fold_left f' l m) in *. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H6. -rewrite <-H5 in H1. rewrite H7 in H1. -inversion H6; subst; clear H6. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H1). elim H3; auto. -generalize (VertexSet.remove_3 H4). intro. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -intuition. -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. -apply IHl. rewrite <-H7. rewrite <-H5 in H1. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H4. auto. auto. - -auto. -auto. -Qed. - -Lemma merge_conflicts_aux_2 : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -~Vertex.eq y (fst_ext e) -> -~Vertex.eq x (fst_ext e) -> -VertexSet.In x (adj_set y (pmap g)). - -Proof. -intros. -unfold map_merge in H1. -set (m := pmap g) in *. -set (f := fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) - m') in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) m)) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) m)))) in *. -set (s' := (adj_set (snd_ext e) m)) in *. -unfold adj_set in H1. -destruct (Vertex.eq_dec y (snd_ext e)). -rewrite MapFacts.remove_eq_o in H1. elim (VertexSet.empty_1 H1). -intuition. -rewrite MapFacts.remove_neq_o in H1. rewrite MapFacts.add_neq_o in H1. -rewrite VertexSet.fold_1 in H1. set (f' := fun a e => f e a) in *. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). simpl in H1. -assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H4 y). clear H4. intro H4. simpl in H1. inversion H4; subst; clear H4. -rewrite <-H6 in H1. elim (VertexSet.empty_1 H1). -set (tmp := fold_left f' l m) in *. -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H6. -rewrite <-H5 in H1. rewrite H7 in H1. -inversion H6; subst; clear H6. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H1). elim H3; auto. -generalize (VertexSet.remove_3 H4). intro. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -intuition. -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. -apply IHl. rewrite <-H7. rewrite <-H5 in H1. assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H4. auto. auto. - -auto. -auto. -Qed. - -Lemma merge_conflicts : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) -> -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -~Vertex.eq y (fst_ext e) -> -~Vertex.eq x (fst_ext e) -> -False. - -Proof. -intros. -apply (simple_graph g x y). -split. -apply (merge_conflicts_aux_1 e g H H0 x y H1 H3 H4). -apply (merge_conflicts_aux_2 e g H H0 x y H2 H3 H4). -Qed. - -Lemma resolve_conflicts_map_3 : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (imap_merge e g)) /\ -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) - --> False. - -Proof. -intros x y e g Haff Hin H. destruct H. -unfold resolve_conflicts in H0. -set (f := (fun (x : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x - (VertexSet.remove (fst_ext e) - (adj_set x (map_merge e (pmap g)))) m)) in *. -set (s := (VertexSet.diff (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (s' := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH s' y). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H0. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H0. rewrite MapFacts.add_eq_o in H0. -elim (VertexSet.diff_2 H0). -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. - -assert (InA Vertex.eq y nil). -apply HH. -unfold s'. apply VertexSet.inter_3. -cut (Vertex.eq x (fst_ext e)). intro. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym H1)). -apply sym_merge; auto. -destruct (Vertex.eq_dec x (fst_ext e)). -intuition. -apply False_ind. apply (merge_conflicts e g Haff Hin x y); auto. - -destruct (Vertex.eq_dec x (fst_ext e)). -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply (sym_imap_merge_map e g Haff Hin). assumption. -apply False_ind. -apply (merge_conflicts e g Haff Hin x y); auto. inversion H1. -auto. - -set (m := map_merge e (pmap g)) in *. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 y). clear H1. simpl in H0. intro H1. inversion H1; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H4. congruence. -intuition. -rewrite MapFacts.add_neq_o in H4. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H0. rewrite MapFacts.add_eq_o in H0. -unfold adj_set in IHl. rewrite MapFacts.add_eq_o in IHl. -apply IHl. assumption. intros. -generalize (HH H1). intro H2. inversion H2; subst. -elim (n H6). auto. -intuition. -intuition. - -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. -rewrite <-H3 in H0. elim (VertexSet.empty_1 H0). -auto. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H3. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H0. -rewrite MapFacts.add_eq_o in H0. -elim (VertexSet.diff_2 H0). -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -intuition. -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. -rewrite <-H2 in H0. -inversion H3; subst; clear H3. -rewrite H4 in H0. clear H4. -destruct (Vertex.eq_dec x (fst_ext e)). -elim (VertexSet.remove_1 (Vertex.eq_sym e1) H0). -apply (merge_conflicts e g Haff Hin x y); auto. -unfold adj_set. rewrite (MapFacts.find_o _ e0). apply (VertexSet.remove_3 H0). -auto. -intuition. - -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in IHl. -unfold adj_set in H0. -rewrite MapFacts.add_eq_o in H0. -apply IHl. assumption. intro. -generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -assumption. -intuition. -intuition. -rewrite MapFacts.add_neq_o in IHl. rewrite <-H3 in IHl. -unfold adj_set in H0. rewrite MapFacts.add_neq_o in H0. -rewrite <-H2 in H0. -apply IHl. rewrite <-H4. assumption. -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -assumption. -auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -Qed. - -Lemma simple_graph_merge_map : forall e g, -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (imap_merge e g)) /\ -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -False. - -Proof. -exact (fun e g H H0 x y H1 => - (resolve_conflicts_map_3 x y e g H H0 H1)). -Qed. - -Lemma sym_map_merge_pmap : forall e g x y, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -VertexSet.In y (adj_set x (map_merge e (pmap g))). - -Proof. -intros e g x y Haff Hin H. -apply sym_map_merge_map with (g:=g); auto. -apply (sym_pmap g). -intros. -apply (not_eq_extremities_map_merge a b e (pmap g)). -intros. apply (not_eq_extremities g). right. assumption. assumption. - -assert (~Vertex.eq y (snd_ext e)) as Hy. -intro. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -apply (pmap_merge_domain_1 y _ _ Hin). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. rewrite H1 in H. elim (VertexSet.empty_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). - -intro. -unfold map_merge in H. - -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y (pmap g)))) - m')) in *. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) - (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) - (adj_set (snd_ext e) (pmap g))))) in *. -set (s' := adj_set (snd_ext e) (pmap g)) in *. - -unfold adj_set in H. -rewrite MapFacts.remove_neq_o in H. -destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o in H. -unfold s in H. -destruct (VertexSet.union_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold s' in H1. - -elim (not_eq_extremities g x (snd_ext e)). -right. assumption. -assumption. -intuition. -rewrite MapFacts.add_neq_o in H. - -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 (VertexSet.remove (fst_ext e) s') y). clear H1. intro H1. -induction (VertexSet.elements (VertexSet.remove (fst_ext e) s')). -simpl in H. -unfold s' in H1. -assert (InA Vertex.eq y nil). -apply H1. -apply VertexSet.remove_2. auto. -apply (sym_pmap g). rewrite <-H0. assumption. inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) (pmap g)) (f' (fold_left f' l (pmap g)) a)). intro. -generalize (H2 y). clear H2. intro H2. simpl in H. inversion H2; clear H2. -rewrite <-H4 in *. -elim (VertexSet.empty_1 H). -set (tmp := fold_left f' l (pmap g)) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. -rewrite <-H3 in *. clear H3. inversion H4. subst. clear H4. -rewrite H5 in H. clear H5. -destruct (proj1 (Props.Dec.F.add_iff _ _ _) H). -elim (In_graph_edge_diff_ext _ _ Hin). -apply Vertex.eq_trans with (y := x); auto. -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H2). -intuition. -rewrite MapFacts.add_neq_o in H4. rewrite <-H4 in IHl. -apply IHl. -rewrite <-H5. rewrite <-H3 in H. assumption. -intro. generalize (H1 H2). clear H2. intro H2. -inversion H2; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -auto. - -intro. -assert (VertexSet.In y (VertexSet.remove (snd_ext e) (V g))). -apply (pmap_merge_domain_1 y _ _ Hin). -rewrite MapFacts.in_find_iff. intro. -unfold adj_set in H. rewrite H1 in H. elim (VertexSet.empty_1 H). -elim (VertexSet.remove_1 (Vertex.eq_sym H0) H1). -Qed. - -Lemma sym_resolve : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) -> -VertexSet.In y (adj_set x (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))). - -Proof. -unfold resolve_conflicts; intros x y e g p q H. -set (f := (fun (x0 : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x0 - (VertexSet.remove (fst_ext e) - (adj_set x0 (map_merge e (pmap g)))) m)) in *. -set (s := (VertexSet.diff (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (s' := (VertexSet.inter (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (m := map_merge e (pmap g)) in *. -rewrite VertexSet.fold_1 in *. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H0 s' y). clear H0. intro HH. -generalize VertexSet.elements_2. intro. -generalize (H0 s'). clear H0. intro HHH. -induction (VertexSet.elements s'). simpl in *. -destruct (Vertex.eq_dec x (fst_ext e)). -unfold adj_set. rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -generalize (VertexSet.diff_1 H). intro H0. -rewrite e0 in H0. -unfold m in H0. unfold map_merge in H0. -unfold adj_set in H0. -rewrite MapFacts.remove_neq_o in H0. -rewrite MapFacts.add_eq_o in H0. -fold (adj_set (fst_ext e) (pmap g)) in H0. -fold (adj_set (snd_ext e) (pmap g)) in H0. -destruct (VertexSet.union_1 H0). -generalize (VertexSet.remove_3 H1). intro. -elim (not_eq_extremities g (fst_ext e) (fst_ext e)). right. auto. -intuition. -elim (VertexSet.remove_1 (Vertex.eq_refl _) H1). -intuition. -apply (In_graph_edge_diff_ext _ _ q). intuition. -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -fold (adj_set y m) in H. -destruct (Props.In_dec y (adj_set (fst_ext e) (imap_merge e g))). -assert (InA Vertex.eq y nil). -apply HH. apply VertexSet.inter_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply (sym_map_merge_pmap e g _ _ p q H). assumption. inversion H0. -apply VertexSet.diff_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply (sym_map_merge_pmap e g _ _ p q H). assumption. -auto. -intuition. - -unfold adj_set. rewrite MapFacts.add_neq_o. -fold (adj_set x m). -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -apply (sym_map_merge_pmap e g _ _ p q). -unfold adj_set. rewrite (MapFacts.find_o _ e0). -apply (VertexSet.diff_1 H). -intuition. -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -apply (sym_map_merge_pmap e g _ _ p q H). -auto. -auto. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -destruct (Vertex.eq_dec x (fst_ext e)). -unfold adj_set. rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -rewrite e1. rewrite <-e0. assumption. -intuition. -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -generalize (H0 y). simpl in H. intro H1. inversion H1; clear H1. -rewrite <-H3 in H. -elim (VertexSet.empty_1 H). -rewrite <-H2 in H. clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H3. -inversion H3; subst; clear H3. -rewrite H4 in H. elim (VertexSet.remove_1 (Vertex.eq_sym e0) H). -intuition. -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. - rewrite MapFacts.add_eq_o in IHl. -rewrite <-H3 in IHl. -apply IHl. rewrite <-H4. assumption. -intros. generalize (HH H1). intro. -inversion H2; subst. -elim (n0 H6). -assumption. -auto. -intuition. -auto. -auto. -auto. -intuition. - -unfold adj_set. rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -generalize (H0 x). intro H1. inversion H1; subst; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -destruct (Vertex.eq_dec y a). -assert (VertexSet.In y s'). apply HHH. -left. auto. -rewrite e0 in H1. -unfold s' in H1. -elim (not_eq_extremities_merge_map e g p (fst_ext e) (fst_ext e)). -left. apply (VertexSet.inter_2 H1). -intuition. -rewrite MapFacts.add_neq_o in H4. -unfold adj_set in IHl. rewrite MapFacts.add_eq_o in IHl. - rewrite MapFacts.add_neq_o in IHl. -rewrite <-H4 in IHl. -assert (VertexSet.In y VertexSet.empty). -apply IHl. assumption. -intro. generalize (HH H1). intro. -inversion H2; subst. -elim (n1 H6). auto. -intros. apply HHH. right. auto. -elim (VertexSet.empty_1 H1). -auto. -intuition. -auto. - -unfold adj_set in IHl. rewrite MapFacts.add_eq_o in IHl. - rewrite MapFacts.add_neq_o in IHl. -simpl. rewrite <-H2. rewrite H4. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -inversion H3; subst; clear H3. -assert (VertexSet.In x s'). -apply HHH. left. auto. -elim (VertexSet.diff_2 H (VertexSet.inter_2 H1)). intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl. -destruct (Vertex.eq_dec y a). -assert (VertexSet.In y s'). apply HHH. left. intuition. -unfold s' in H1. -elim (not_eq_extremities_merge_map e g p (fst_ext e) (fst_ext e)). -left. rewrite e0 in H1. apply (VertexSet.inter_2 H1). intuition. - -apply IHl. -assumption. -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n1 H7). -auto. - -intros. apply HHH. right. auto. -auto. -auto. -intuition. -intuition. - -unfold adj_set in H. rewrite MapFacts.add_neq_o in H. -simpl in H. -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. - rewrite MapFacts.add_neq_o in IHl. -generalize (H0 y). intro H1. inversion H1; clear H1. -rewrite <-H3 in H. elim (VertexSet.empty_1 H). -rewrite <-H2 in H. -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). rewrite MapFacts.add_eq_o in H3. -inversion H3; subst; clear H3. -generalize H. clear H. intro H. rewrite H4 in H. -generalize (VertexSet.remove_3 H). clear H. intro H. -unfold adj_set in H. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)) in H. -fold (adj_set y m) in H. -generalize (sym_map_merge_pmap e g _ _ p q H). clear H. intro H. -fold m in H. -clear IHl H0 H2 HH HHH. - -set (l' := a :: l) in *. -induction l'. simpl. assumption. - -(* - unfold f'. unfold f. -destruct (Vertex.eq_dec x a). rewrite MapFacts.add_eq_o. -apply VertexSet.remove_2; auto. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). auto. -auto. -rewrite MapFacts.add_neq_o. assumption. -auto. -*) - -cut (EqualSetMap (fold_left f' (a0 :: l') m) (f' (fold_left f' l' m) a0)). intro. -generalize (H0 x). clear H0. intro H0. simpl. inversion H0; clear H0. -set (tmp' := fold_left f' l' m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H3. congruence. intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl'. elim (VertexSet.empty_1 IHl'). -auto. -set (tmp' := fold_left f' l' m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H2. inversion H2; subst; clear H2. -rewrite H3. apply VertexSet.remove_2. auto. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -intuition. -rewrite MapFacts.add_neq_o in H2. -rewrite <-H2 in IHl'. -rewrite H3. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s1); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. intuition. - -intuition. -rewrite MapFacts.add_neq_o in H3. -rewrite <-H3 in IHl. - -generalize (H0 x). intro. inversion H1; clear H1. -unfold f' in H7. unfold f in H7. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H7. congruence. intuition. -rewrite MapFacts.add_neq_o in H7. -rewrite <-H7 in IHl. simpl. rewrite <-H6. -apply IHl. -rewrite <-H4. auto. - -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n1 H9). -auto. - -intros. apply HHH. right. auto. -auto. - -unfold f' in H6. unfold f in H6. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H6. -inversion H6; subst; clear H6. -simpl. rewrite <-H5. -rewrite H7. -apply VertexSet.remove_2. auto. - -case_eq (VertexMap.find x tmp); intros. -rewrite H1 in IHl. -assert (VertexSet.In y t0). -apply IHl. rewrite <-H4. assumption. - -intro. generalize (HH H6). intro. -inversion H8; subst. -elim (n1 H10). -auto. - -intros. apply HHH. right. auto. -clear H HH HHH IHl H0 H4 H2 H3 H5 H7. -assert (eq_set_option (Some t0) (VertexMap.find x tmp)). -rewrite H1. constructor. apply VertexSet.eq_refl. clear H1. -unfold tmp in H. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -fold (adj_set x m). -induction l. simpl in H. inversion H. -unfold adj_set. rewrite <-H1. rewrite <-H2. auto. - -cut (EqualSetMap (fold_left f' (a0 :: l) m) (f' (fold_left f' l m) a0)). intro. -generalize (H0 x). clear H0. intro H0. simpl in H. inversion H0; subst. -rewrite <-H2 in H. inversion H. -set (tmp' := fold_left f' l m) in *. -unfold f' in H2. unfold f in H2. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H2. -inversion H; subst. rewrite <-H1 in H5. -inversion H5; subst; clear H5. -rewrite H7 in H6. rewrite H3 in H6. inversion H2; subst; clear H2 H3 H7. -unfold adj_set. rewrite (MapFacts.find_o _ e1). apply (VertexSet.remove_3 H6). intuition. -rewrite MapFacts.add_neq_o in H2. rewrite <-H2 in IHl. -apply IHl. constructor. rewrite <-H1 in H. inversion H; subst. -rewrite H7. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e2)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s2); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. - -rewrite H1 in IHl. -assert (VertexSet.In y VertexSet.empty). -apply IHl. -rewrite <-H4. auto. -intro. generalize (HH H6). intro. -inversion H8; subst. -elim (n1 H10). -auto. - -intros. apply HHH. right. auto. -elim (VertexSet.empty_1 H6). intuition. - -rewrite MapFacts.add_neq_o in H6. -rewrite <-H6 in IHl. simpl. rewrite <-H5. -rewrite H7. apply IHl. -rewrite <-H4. auto. -intro. generalize (HH H1). intro. -inversion H8; subst. -elim (n1 H10). -auto. - -intros. apply HHH. right. auto. -auto. -auto. -auto. -auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. auto. auto. -Qed. - -Lemma sym_pmap_merge_map e g : -aff_edge e -> -In_graph_edge e g -> -forall x y, -VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) -> -VertexSet.In y (adj_set x (pmap_merge e g (imap_merge e g))). - -Proof. -intros. apply sym_resolve; assumption. -Qed. - -Definition merge e g - (q : In_graph_edge e g) - (p : aff_edge e) := - let im := imap_merge e g in - let pm := pmap_merge e g im in - Make_Graph (VertexSet.remove (snd_ext e) (V g)) - im - pm - (extremities_in_merge_imap e g q) - (extremities_in_merge_pmap e g p q) - (simple_graph_merge_map e g p q) - (sym_imap_merge_map e g p q) - (sym_pmap_merge_map e g p q) - (not_eq_extremities_merge_map e g p). - -Lemma In_graph_dec : forall v g, -{In_graph v g} + {~In_graph v g}. - -Proof. -exact (fun v g => Props.In_dec v (V g)). -Qed. - -Definition pmap_delete_preferences v g := -let pm := pmap g in -let m := VertexMap.add v VertexSet.empty pm in -VertexSet.fold - (fun y m' => VertexMap.add y (VertexSet.remove v (adj_set y pm)) m') - (adj_set v pm) - m. - -Lemma delete_preference_sub : forall x v g, -VertexSet.Subset (adj_set x (pmap_delete_preferences v g)) - (adj_set x (pmap g)). - -Proof. -unfold VertexSet.Subset;intros. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. - -generalize VertexSet.elements_2. -intro H0. generalize (H0 (adj_set v (pmap g))). clear H0. intro H0. -induction (VertexSet.elements (adj_set v (pmap g))); intros. -simpl in H. -unfold adj_set in H. -destruct (Vertex.eq_dec v x). -rewrite InterfFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). -assumption. -rewrite InterfFacts.add_neq_o in H. auto. -assumption. -cut (EqualSetMap - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) (a0 :: l) - (VertexMap.add v VertexSet.empty (pmap g))) - (VertexMap.add a0 - (VertexSet.remove v - (adj_set a0 (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) l - (VertexMap.add v VertexSet.empty (pmap g))))). - -intro. -assert (VertexSet.In a (adj_set x - (VertexMap.add a0 - (VertexSet.remove v - (adj_set a0 (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) l - (VertexMap.add v VertexSet.empty (pmap g)))))). -unfold EqualSetMap in H1. generalize (H1 x). intro H2. -inversion H2. -unfold adj_set. unfold adj_set in H5. rewrite <-H5. -unfold adj_set in *. simpl in H. rewrite <-H4 in H. -assumption. -unfold adj_set in *. rewrite <-H4. -unfold adj_set in *. simpl in H. rewrite <-H3 in H. -rewrite <-H5. assumption. -generalize (H1 x). clear H1. intro H1. inversion H1. -simpl in H. unfold adj_set in H. unfold adj_set in H4. rewrite <-H4 in H. -elim (VertexSet.empty_1 H). -clear H1. -destruct (Vertex.eq_dec x a0). -unfold adj_set in H2. rewrite InterfFacts.add_eq_o in H2. -generalize (VertexSet.remove_3 H2). unfold adj_set. -rewrite (InterfFacts.find_o _ e). auto. intuition. -apply IHl. -rewrite InterfFacts.add_neq_o in H4. -unfold adj_set in *. rewrite <-H4. -rewrite <-H5. simpl in H. rewrite <-H3 in *. -assumption. -auto. -intuition. - -apply fold_left_assoc_map. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e). intro HH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HH). apply VertexSet.eq_refl. intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply EqualSetMap_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a1). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply (H1 x0). -auto. -auto. -Qed. - -Lemma in_pmap_delete_in : forall x v g, -In_graph v g -> -VertexMap.In x (pmap_delete_preferences v g) -> -VertexMap.In x (pmap g). - -Proof. -intros x v g H1 H. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. - -generalize VertexSet.elements_2. intro HH. -generalize (HH (adj_set v (pmap g))). clear HH. intro HH. - -induction (VertexSet.elements (adj_set v (pmap g))); intros. -simpl in H. -destruct (proj1 (InterfFacts.add_in_iff _ _ _ _) H). -apply (proj2 (extremities_pmap g x)). -rewrite <-H0. assumption. -assumption. - -cut (EqualSetMap - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) (a :: l) - (VertexMap.add v VertexSet.empty (pmap g))) - (VertexMap.add a - (VertexSet.remove v - (adj_set a (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) l - (VertexMap.add v VertexSet.empty (pmap g))))). intro. -unfold EqualSetMap in H0. - -generalize (proj1 (InterfFacts.in_find_iff _ _) H). clear H. intro H. -generalize (H0 x). intro H2. -inversion H2. -unfold adj_set in *. simpl in H. rewrite <-H4 in H. -elim H. auto. -clear H2. -destruct (Vertex.eq_dec x a). -rewrite InterfFacts.add_eq_o in H4. -apply (proj2 (InterfFacts.in_find_iff _ _)). -assert (VertexSet.In v (adj_set x (pmap g))). -apply (sym_pmap g). -apply HH. left. auto. -unfold adj_set in H2. -destruct (VertexMap.find x (pmap g)). -intro Helim. inversion Helim. -elim (VertexSet.empty_1 H2). intuition. -rewrite InterfFacts.add_neq_o in H4. -apply IHl. -apply (proj2 (InterfFacts.in_find_iff _ _)). -rewrite <-H4. intro Helim. inversion Helim. -intuition. -auto. - -apply fold_left_assoc_map. -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e). intro HHH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HHH). apply VertexSet.eq_refl. intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply EqualSetMap_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply (H0 x0). -auto. -auto. -Qed. - -Lemma extremities_in_delete_preferences_imap (v : Vertex.t) g : -forall x, -VertexMap.In x (imap g) <-> VertexSet.In x (V g). - -Proof. -exact (fun v g => extremities_imap g). -Qed. - -Lemma extremities_in_delete_preferences_pmap v g : -In_graph v g -> -(forall x, -VertexMap.In x (pmap_delete_preferences v g) <-> -VertexSet.In x (V g)). - -Proof. -split; intros. -apply (proj1 (extremities_pmap g x)). -apply in_pmap_delete_in with (v:=v); auto. - -unfold pmap_delete_preferences. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y (VertexSet.remove v (adj_set y (pmap g))) m')) in *. -set (m := VertexMap.add v VertexSet.empty (pmap g)). -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set v (pmap g))). simpl. -unfold m. -rewrite MapFacts.in_find_iff. -destruct (Vertex.eq_dec x v). -rewrite MapFacts.add_eq_o. -congruence. intuition. -rewrite MapFacts.add_neq_o. -generalize (proj2 (extremities_pmap g x) H0). -rewrite MapFacts.in_find_iff. auto. -auto. -set (h := (fun (a0 : VertexMap.t VertexSet.t) (e : VertexSet.elt) => f e a0)) in *. - -cut (EqualSetMap (fold_left h (a :: l) m) (h (fold_left h l m) a)). intro. -generalize (H1 x). clear H1. intro H1. -inversion H1. -set (tmp := fold_left h l m) in *. -unfold h in H4. -unfold f in H4. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H4. -congruence. intuition. -rewrite MapFacts.add_neq_o in H4. -rewrite MapFacts.in_find_iff in IHl. rewrite <-H4 in IHl. -congruence. intuition. -simpl. rewrite MapFacts.in_find_iff. rewrite <-H2. congruence. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold h. unfold f. intros. -destruct (Vertex.eq_dec x0 z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e). intro HHH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HHH). apply VertexSet.eq_refl. -intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -intros. -unfold EqualSetMap. unfold h. unfold f. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply H1. auto. auto. -Qed. - -(* -Lemma extremities_in_delete_preferences v g : -forall x, -VertexMap.In x (imap g) \/ -VertexMap.In x (pmap_delete_preferences v g) <-> -VertexSet.In x (V g). - -Proof. -split; intros. -destruct (Vertex.eq_dec v x). -apply (proj1 (extremities g x)). -destruct H;[left|right]. -assumption. -apply in_pmap_delete_in with (v:=v); auto. - -Qed. -*) - -Lemma simple_graph_delete_preferences v g : forall x y, -VertexSet.In x (adj_set y (imap g)) /\ -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> False. - -Proof. -intros. -destruct H. -generalize (delete_preference_sub _ _ _ _ H0). clear H0. intro H0. -apply (simple_graph g x y). intuition. -Qed. - -Lemma sym_imap_delete_preferences (v : Vertex.t) g : -forall x y, -VertexSet.In x (adj_set y (imap g)) -> -VertexSet.In y (adj_set x (imap g)). - -Proof. -exact (fun v g => sym_imap g). -Qed. - -Lemma in_adj_delete_preference_not_eq_1 : forall x y v g, -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -~Vertex.eq y v. - -Proof. -intros. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a)) in *. -set (l := VertexSet.elements (adj_set v (pmap g))) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH (adj_set v (pmap g))). clear HH. intro HH. -fold l in HH. -assert (forall z, In z l -> ~Vertex.eq v z) as Hneq. -clear H. -intros. intro H0. -induction l. inversion H. -simpl in H. destruct H. subst. -assert (VertexSet.In v (adj_set v (pmap g))). -apply HH. left. auto. -elim (not_eq_extremities g v v). -right. assumption. -auto. -apply IHl. -intros. apply HH. auto. -assumption. - -induction l. simpl in H. -unfold adj_set in H. intro Helim. -rewrite MapFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). intuition. -set (s := VertexMap.add v VertexSet.empty (pmap g)) in *. - -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -intro Helim. -cut (VertexSet.In x (adj_set y (f (fold_left f l s) a))). -set (tmp := fold_left f l s) in *. -intro H1. -unfold f in H1. -generalize (H0 x). clear H0. intro H0. -unfold adj_set in H1. inversion H0. -rewrite <-H4 in *. -destruct (Vertex.eq_dec y a). -elim (Hneq a). left. auto. -apply (Vertex.eq_trans (Vertex.eq_sym Helim) e). -rewrite MapFacts.add_neq_o in H1. -apply IHl. -assumption. -intros. apply HH. auto. -intros. apply Hneq. right. auto. -assumption. -auto. - -rewrite <-H3 in *. -destruct (Vertex.eq_dec y a). -elim (Hneq a). left. auto. -apply (Vertex.eq_trans (Vertex.eq_sym Helim) e). -rewrite MapFacts.add_neq_o in H1. -apply IHl. -assumption. -intros. apply HH. auto. -intros. apply Hneq. right. auto. -assumption. -auto. - -clear HH Hneq IHl. -unfold adj_set in H. unfold adj_set. -generalize (H0 y). clear H0. intro H0. -simpl in H. -inversion H0; subst. -rewrite <-H3 in *. -simpl in H. rewrite <-H2 in *. -elim (VertexSet.empty_1 H). -rewrite <-H1 in *. -rewrite <-H2 in *. -rewrite <-H3. assumption. -Qed. - -Lemma in_adj_delete_preference_not_eq_2 : forall x y v g, -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -~Vertex.eq x v. - -Proof. -intros. intro Helim. generalize H. intro Hcopy. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. -set (f:= (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a)) in *. -generalize VertexSet.elements_1. intro. -generalize (H0 (adj_set v (pmap g)) y). clear H0. intro HH. -set (l := VertexSet.elements (adj_set v (pmap g))) in *. -induction l. simpl in H. -unfold adj_set in H. -rewrite MapFacts.add_neq_o in H. -generalize (sym_pmap g). intro Hsym. -generalize (Hsym _ _ H). intro H0. -unfold adj_set in H0. -rewrite (MapFacts.find_o _ Helim) in H0. -generalize (HH H0). intro H1. inversion H1. -generalize (in_adj_delete_preference_not_eq_1 x y v g Hcopy). auto. -set (s := VertexMap.add v VertexSet.empty (pmap g)) in *. - -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -clear Hcopy. -cut (VertexSet.In x (adj_set y (f (fold_left f l s) a))). -set (tmp := fold_left f l s) in *. -intro H1. -unfold f in H1. unfold adj_set in H1. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H1. -elim (VertexSet.remove_1 (Vertex.eq_sym Helim) H1). intuition. -rewrite MapFacts.add_neq_o in H1. -apply IHl. -assumption. -intro. generalize (HH H2). intro H3. -inversion H3; subst. -elim (n H5). -assumption. -auto. - -unfold adj_set. -generalize (H0 y). clear H0. intro H0. -inversion H0; subst. -simpl in H. unfold adj_set in H. rewrite <-H2 in H. -assumption. -simpl in H. unfold adj_set in H. rewrite <-H1 in H. -rewrite <-H3. assumption. -Qed. - -Lemma sym_pmap_delete_preferences (v : Vertex.t) g : -forall x y, -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -VertexSet.In y (adj_set x (pmap_delete_preferences v g)). - -Proof. -intros. -generalize (in_adj_delete_preference_not_eq_1 x y v g H). intro Hneqy. -generalize (in_adj_delete_preference_not_eq_2 x y v g H). intro Hneqx. -generalize (delete_preference_sub y v g x H). intro HH. -generalize (sym_pmap g _ _ HH). clear HH. intro HH. -unfold pmap_delete_preferences in *. -rewrite VertexSet.fold_1 in *. -set (f := fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove v (adj_set e (pmap g))) a) in *. -induction (VertexSet.elements (adj_set v (pmap g))). -simpl in *. -destruct (Vertex.eq_dec x v). -destruct (Vertex.eq_dec y v). -unfold adj_set in H. rewrite MapFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). -case_eq (VertexMap.find y (VertexMap.add v VertexSet.empty (pmap g))). intros. -rewrite H0 in H. -rewrite MapFacts.add_eq_o in H0. inversion H0. subst. -elim (VertexSet.empty_1 H). intuition. -rewrite MapFacts.add_eq_o. congruence. intuition. - -elim (Hneqx e). - -unfold adj_set. rewrite MapFacts.add_neq_o. assumption. intuition. - -set (s := VertexMap.add v VertexSet.empty (pmap g)) in *. -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -unfold EqualSetMap in H0. -unfold adj_set. -case_eq (VertexMap.find x (fold_left f (a :: l) s)); intros. -generalize (H0 x). intro HH0. -rewrite H1 in HH0. inversion HH0. subst. -set (tmp := fold_left f l s) in *. -unfold adj_set in H. -unfold f in H3. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H3. -inversion H3. subst. clear H3. -rewrite H4. apply VertexSet.remove_2. -auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). assumption. -intuition. - -rewrite MapFacts.add_neq_o in H3. -rewrite H4. -cut (VertexSet.In y (adj_set x tmp)). -intro. -unfold adj_set in H2. -rewrite <-H3 in H2. assumption. -apply IHl. -assert (VertexSet.In x (adj_set y (pmap g)) -> VertexSet.In x (adj_set y tmp)). -clear IHl H0 H1 HH0 H4 H3 HH H. -intros. -unfold tmp. -induction l. simpl. unfold s. -unfold adj_set. -rewrite MapFacts.add_neq_o. assumption. -auto. - -assert (EqualSetMap (fold_left f (a0 :: l) s) (f (fold_left f l s) a0)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -generalize (H0 y). clear H0. intro H0. -inversion H0; subst. -unfold adj_set. simpl. rewrite <-H2. -set (tmp' := fold_left f l s) in *. -unfold f in H3. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H3. inversion H3. -intuition. -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. rewrite <-H3 in IHl. -assumption. -auto. -unfold adj_set. simpl. rewrite <-H1. -rewrite H3. -set (tmp' := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec y a0). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -apply VertexSet.remove_2. -auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). -assumption. -intuition. -rewrite MapFacts.add_neq_o in H2. -assert (VertexSet.In x (adj_set y tmp')). -apply IHl. -unfold adj_set in H4. -rewrite <-H2 in H4. assumption. -auto. -apply H2. -apply (sym_pmap g). assumption. -auto. - -generalize (H0 x). clear H0. intro H0. -rewrite H1 in H0. inversion H0. subst. -set (tmp := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec x a). -rewrite MapFacts.add_eq_o in H2. -inversion H2. -intuition. -rewrite MapFacts.add_neq_o in H2. - -assert (VertexSet.In y (adj_set x (pmap g)) -> VertexSet.In y (adj_set x tmp)). -clear IHl H H0 HH H2 H1. -intros. -unfold tmp. -induction l. simpl. unfold s. -unfold adj_set. -rewrite MapFacts.add_neq_o. assumption. -intuition. - -assert (EqualSetMap (fold_left f (a0 :: l) s) (f (fold_left f l s) a0)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e) e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x0 a1). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -generalize (H0 x). clear H0. intro H0. -inversion H0; subst. -unfold adj_set. simpl. rewrite <-H2. -set (tmp' := fold_left f l s) in *. -unfold f in H3. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H3. inversion H3. -intuition. -rewrite MapFacts.add_neq_o in H3. -unfold adj_set in IHl. rewrite <-H3 in IHl. -assumption. -auto. -unfold adj_set. simpl. rewrite <-H1. -rewrite H3. -set (tmp' := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o in H2. -inversion H2. subst. clear H2. -apply VertexSet.remove_2. -auto. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e)). -assumption. -intuition. -rewrite MapFacts.add_neq_o in H2. -assert (VertexSet.In y (adj_set x tmp')). -apply IHl. -unfold adj_set in H4. -rewrite <-H2 in H4. assumption. -auto. -assert (VertexSet.In y (adj_set x tmp)). -apply H3. assumption. -unfold adj_set in H4. -rewrite <-H2 in H4. -assumption. -auto. -Qed. - -Lemma not_eq_extremities_delete_preferences v g : -forall x y, -VertexSet.In x (adj_set y (imap g)) \/ -VertexSet.In x (adj_set y (pmap_delete_preferences v g)) -> -~Vertex.eq x y. - -Proof. -intros. -apply (not_eq_extremities g). -destruct H;[left|right]. -assumption. -apply (delete_preference_sub _ _ _ _ H). -Qed. - -Definition delete_preference_edges v g H := -Make_Graph (V g) - (imap g) - (pmap_delete_preferences v g) - (extremities_in_delete_preferences_imap v g) - (extremities_in_delete_preferences_pmap v g H) - (simple_graph_delete_preferences v g) - (sym_imap_delete_preferences v g) - (sym_pmap_delete_preferences v g) - (not_eq_extremities_delete_preferences v g). - -Definition Prefere x y g := -exists w, In_graph_edge (x,y,Some w) g. - -Definition Interfere x y g := -In_graph_edge (x,y,None) g. - -Definition precolored g := -VertexSet.filter (fun v => is_precolored v g) (V g). - -Parameter mreg_ext : forall x y, -Vertex.eq x y -> is_mreg x = is_mreg y. - -Lemma mem_ext : forall x y s, -Vertex.eq x y -> VertexSet.mem x s = VertexSet.mem y s. - -Proof. -intros. -case_eq (VertexSet.mem x s); intros. -generalize (VertexSet.mem_2 H0). intro H1. -rewrite H in H1. generalize (VertexSet.mem_1 H1). auto. -case_eq (VertexSet.mem y s); intros. -generalize (VertexSet.mem_2 H1). intro H2. -rewrite <-H in H2. generalize (VertexSet.mem_1 H2). intro H3. -rewrite H3 in H0. inversion H0. -reflexivity. -Qed. - -Lemma compat_bool_is_precolored : forall g, -compat_bool Vertex.eq (fun x => is_precolored x g). - -Proof. -unfold compat_bool. intros. -unfold is_precolored. -apply mreg_ext. auto. -Qed. - -(* Equivalence of predicate and function *) -Lemma precolored_equiv : forall x g, -VertexSet.In x (precolored g) <-> (is_precolored x g = true /\ In_graph x g). - -Proof. -split; intros. -split. -apply (VertexSet.filter_2 (compat_bool_is_precolored _) H). -apply (VertexSet.filter_1 (compat_bool_is_precolored _) H). -apply VertexSet.filter_3. apply compat_bool_is_precolored. -unfold is_precolored in H. -case_eq (VertexSet.mem x (V g)); intros. -apply VertexSet.mem_2. assumption. intuition. intuition. -Qed. - -Lemma In_graph_aff_edge_in_AE : forall e g, -EdgeSet.In e (AE g) <-> aff_edge e /\ In_graph_edge e g. - -Proof. -intros e g. split; intro H. -split. -unfold aff_edge. exists N0. exact (AE_weights g e H). -left. assumption. -destruct H as [H H0]. -destruct H0. -assumption. -unfold aff_edge in H. destruct H as [w H]. -rewrite (IE_weights g e H0) in H. inversion H. -Qed. - -Lemma In_graph_interf_edge_in_IE : forall e g, -EdgeSet.In e (IE g) <-> interf_edge e /\ In_graph_edge e g. - -Proof. -intros e g. split; intro H. -split. -exact (IE_weights g e H). -right. assumption. -destruct H as [H H0]. -destruct H0. -unfold interf_edge in H. -rewrite (AE_weights g e H0) in H. inversion H. -assumption. -Qed. - -(* Spec of Prefere *) -Lemma Prefere_1 : forall e g, -aff_edge e -> -In_graph_edge e g -> -Prefere (fst_ext e) (snd_ext e) g. - -Proof. -intros. -unfold Prefere. -destruct H0. -exists N0. rewrite <-(AE_weights g e H0). rewrite <-edge_eq. left. assumption. -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intro H1. -destruct H1 as [H1 _]. unfold interf_edge in H1. unfold aff_edge in H. -rewrite H1 in H. inversion H. inversion H2. -Qed. - -Lemma Prefere_2 : forall x y g, -Prefere x y g -> -In_graph_edge (x,y,Some N0) g. - -Proof. -intros. -unfold Prefere in H. -destruct H. -assert (EdgeSet.In (x,y,Some x0) (AE g)). -rewrite In_graph_aff_edge_in_AE. split. exists x0. auto. auto. -generalize (AE_weights _ _ H0). intro. simpl in H1. rewrite H1 in H. auto. -Qed. - -(* spec of Interfere *) -Lemma Interfere_1 : forall e g, -interf_edge e -> -In_graph_edge e g -> -Interfere (fst_ext e) (snd_ext e) g. - -Proof. -intros. -unfold Interfere. -destruct H0. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H0). intro H1. -destruct H1 as [H1 _]. unfold interf_edge in H. unfold aff_edge in H1. -rewrite H in H1. inversion H1. inversion H2. -rewrite <-(IE_weights g e H0). rewrite <-edge_eq. right. assumption. -Qed. - -Lemma Interfere_2 : forall x y g, -Interfere x y g -> -In_graph_edge (x,y,None) g. - -Proof. -auto. -Qed. - -(* Machine registers are registers *) -Lemma In_precolored : forall g , -VertexSet.Subset (precolored g) (V g). - -Proof. -intros. -unfold VertexSet.Subset; intros. -apply (VertexSet.filter_1 (compat_bool_is_precolored _) H). -Qed. - -(* specification of remove_vertex *) -Lemma In_remove : forall x r g, -In_graph x g -> -~Vertex.eq x r -> -In_graph x (remove_vertex r g). - -Proof. -intros. -unfold remove_vertex. simpl. -apply VertexSet.remove_2; auto. -Qed. - -(* A precolored vertex cannot be removed from the graph *) -Lemma not_in_remove : forall r g, -~In_graph r (remove_vertex r g). - -Proof. -intros. -unfold remove_vertex. -apply VertexSet.remove_1. apply Vertex.eq_refl. -Qed. - -Lemma in_remove_in : forall x r g, -In_graph x (remove_vertex r g) -> -In_graph x g. - -Proof. -intros. -unfold remove_vertex in H. -unfold In_graph in H. simpl in H. -apply (VertexSet.remove_3 H). -Qed. - -Add Morphism In_graph : In_graph_m. - -Proof. -intros. -unfold In_graph. -rewrite H. reflexivity. -Qed. - -(* Probably redundant, TODO get out of interface *) -Lemma precolored_remove_vertex2 : forall x y g, -VertexSet.In x (VertexSet.remove y (precolored g)) <-> -VertexSet.In x (precolored (remove_vertex y g)). - -Proof. -intros. -split; intros. -generalize (VertexSet.remove_3 H). intro H0. -generalize (VertexSet.filter_1 (compat_bool_is_precolored _) H0). intro. -generalize (VertexSet.filter_2 (compat_bool_is_precolored _) H0). clear H0. intro. -apply VertexSet.filter_3. -apply compat_bool_is_precolored. -unfold remove_vertex. simpl. -apply VertexSet.remove_2. intro. apply (VertexSet.remove_1 H2 H). -assumption. -unfold is_precolored. assumption. -generalize (proj1 (precolored_equiv _ _) H). clear H. intro H. destruct H. -apply VertexSet.remove_2. -intro H1. rewrite <-H1 in H0. elim (not_in_remove _ _ H0). -apply (proj2 (precolored_equiv _ _)). -split. -assumption. -apply in_remove_in with (r:=y). assumption. -Qed. - -Lemma In_remove_edge_ : forall e r g, -In_graph_edge e g -> -~incident e r -> -In_graph_edge e (remove_vertex r g). - -Proof. -intros. -destruct H;[left|right]. -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -unfold remove_vertex. -simpl. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap_remove_vertex r g))). -apply imap_remove_1. -intro H1. elim H0. right. auto. -intro H1. elim H0. left. auto. -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g)) H). - -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -unfold remove_vertex. -simpl. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap_remove_vertex r g))). -apply imap_remove_1. -intro H1. elim H0. right. auto. -intro H1. elim H0. left. auto. -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H). -Qed. - -Lemma in_remove_in_edge : forall e r g, -In_graph_edge e (remove_vertex r g) -> -In_graph_edge e g. - -Proof. -intros. -destruct H;[left|right]. -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -unfold remove_vertex in H. -simpl in H. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g))). -apply imap_remove_3 with (r:=r). -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap_remove_vertex r g)) H). - -generalize (IE_weights _ _ H). intro Hw. -unfold IE in *. -unfold remove_vertex in H. -simpl in H. -rewrite (edge_eq e) in *. -simpl in Hw. rewrite Hw. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g))). -apply imap_remove_3 with (r:=r). -rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_imap_remove_vertex r g)) H). -Qed. - -Lemma not_in_remove_edge : forall e r g, -incident e r -> -~In_graph_edge e (remove_vertex r g). - -Proof. -intros. generalize H. intro H0. -destruct (In_graph_edge_dec e g). -assert (In_graph r g) as Hin. -destruct H. -rewrite H. apply (proj1 (In_graph_edge_in_ext _ _ i)). -rewrite H. apply (proj2 (In_graph_edge_in_ext _ _ i)). -intro H1. destruct H1. -generalize (AE_weights _ _ H1). intro Hw. -unfold AE in *. -unfold remove_vertex in H1. -simpl in H1. -rewrite (edge_eq e) in H1. -rewrite Hw in H1. rewrite edge_comm in H1. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap_remove_vertex r g)) H1). -intro H3. -apply (imap_remove_2 _ _ (pmap g) _ H (sym_pmap g) H3). - -generalize (IE_weights _ _ H1). intro Hw. -unfold IE in *. -unfold remove_vertex in H1. -simpl in H1. -rewrite (edge_eq e) in H1. -rewrite Hw in H1. rewrite edge_comm in H1. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap_remove_vertex r g)) H1). -intro H3. -apply (imap_remove_2 _ _ (imap g) _ H (sym_imap g) H3). -intro H1. elim (n (in_remove_in_edge _ _ _ H1)). -Qed. - - -(* spec of merge *) -Lemma In_merge_in : forall g e x p q, -In_graph x (merge e g p q) -> In_graph x g. - -Proof. -intros. -unfold merge in H. -destruct (aff_edge_dec e). -destruct (In_graph_edge_dec e g). -unfold In_graph in H. simpl in H. -apply (VertexSet.remove_3 H). -elim (n p). -elim (n q). -Qed. - -(* the second extremity must not be precolored, - cause it is removed from the graph *) -Lemma merge_1 : forall e g p q, -~In_graph (snd_ext e) (merge e g p q). - -Proof. -intros. -unfold merge. unfold In_graph. simpl. -apply VertexSet.remove_1. apply Vertex.eq_refl. -Qed. - -Lemma merge_2 : forall x e g p q, -~Vertex.eq x (snd_ext e) -> -In_graph x g -> -In_graph x (merge e g p q). - -Proof. -intros. unfold merge. -unfold In_graph. simpl. apply VertexSet.remove_2 with (x:=(snd_ext e)); auto. -Qed. - -Lemma merge_3 : forall e e' g p q, -In_graph_edge e' g -> -interf_edge e' -> -In_graph_edge (redirect (snd_ext e) (fst_ext e) e') (merge e g p q). - -Proof. -intros e e' g p q H0 H1. generalize I. intro H. -assert (EdgeSet.In (fst_ext e, snd_ext e, Some N0) (AE g)) as He. -destruct p. rewrite (edge_eq e) in H2. -generalize (AE_weights _ _ H2). intro. simpl in H3. -rewrite H3 in H2. assumption. -generalize (IE_weights _ _ H2). inversion q. congruence. -assert (VertexSet.In (snd_ext e) (adj_set (fst_ext e) (pmap g))) as Hee. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assumption. -assert (EdgeSet.In (fst_ext e', snd_ext e', None) (IE g)) as He'. -destruct H0. generalize (AE_weights _ _ H0). intro H2. inversion H2. congruence. -rewrite (edge_eq e') in H0. generalize (IE_weights _ _ H0). intro. simpl in H2. -rewrite H2 in H0. assumption. -assert (VertexSet.In (snd_ext e') (adj_set (fst_ext e') (imap g))) as Hee'. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -assumption. -assert (forall a b, VertexSet.In a (adj_set b (imap_merge e g)) -> - VertexSet.In b (adj_set a (imap_merge e g))) as Hsym. -apply sym_imap_merge_map; auto. - -right. -unfold merge. unfold IE. simpl. -unfold redirect. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -rewrite H1. apply (proj2 (edgemap_to_edgeset_charac (imap_merge e g) _ _ None Hsym)). - -unfold imap_merge. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_3. -unfold s'. -apply VertexSet.remove_2. -intro. -elim (simple_graph g (fst_ext e) (snd_ext e)). -split. -rewrite H2. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') (imap g)). -assumption. -apply (sym_pmap g). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') m). -assumption. -intuition. -apply (In_graph_edge_diff_ext _ _ p). - -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -rewrite H1. -apply (proj2 (edgemap_to_edgeset_charac (imap_merge e g) _ _ None Hsym)). -unfold imap_merge. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -induction (VertexSet.elements s'). simpl. -assert (InA Vertex.eq (fst_ext e') nil). -apply HH. -unfold s'. apply VertexSet.remove_2. -intro. -elim (simple_graph g (fst_ext e) (snd_ext e)). -split. -rewrite H2. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') (imap g)). -apply (sym_imap g). assumption. -apply (sym_pmap g). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -apply (sym_imap g). assumption. -inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. -intro. generalize (HH H2). clear HH. intro HH. -inversion HH; subst. -elim (n0 H6). -assumption. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_1. intuition. intuition. -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -intro. generalize (HH H2). clear HH. intro HH. inversion HH; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -intro. -elim (simple_graph g (fst_ext e) (snd_ext e)). -split. -rewrite H2. unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') (imap g)). -apply (sym_imap g). assumption. -apply (sym_pmap g). assumption. - -intuition. -rewrite (edge_eq e'). rewrite H1. -apply (proj2 (edgemap_to_edgeset_charac (imap_merge e g) _ _ None Hsym)). -unfold imap_merge. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_2. -apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -assumption. -intuition. - -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -(* -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -*) -induction (VertexSet.elements s'). simpl. assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_2. apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. - -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -intuition. -Qed. - -Lemma merge_4 : forall e e' g p q, -In_graph_edge e' (merge e g p q) -> -exists a, In_graph_edge a g /\ weak_eq e' (redirect (snd_ext e) (fst_ext e) a) - /\ same_type a e'. - -Proof. -intros e e' g p q H0. generalize I. intro H. -generalize H0. intro Hin. -unfold merge in H0. destruct H0. -generalize (AE_weights _ _ H0). intro Hw. -unfold AE in H0. simpl in H0. -rewrite (edge_eq e') in H0. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map _ _ q p)) H0). -clear H0. intro H0. -generalize (pmap_merge_sub _ _ _ _ H0). clear H0. intro H0. -unfold map_merge in H0. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set in H0. -destruct (Vertex.eq_dec (fst_ext e') (snd_ext e)). -rewrite MapFacts.remove_eq_o in H0. -elim (VertexSet.empty_1 H0). intuition. -rewrite MapFacts.remove_neq_o in H0. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o in H0. -unfold s in H0. -destruct (VertexSet.union_1 H0). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -exists (fst_ext e', snd_ext e', Some N0). split. -left. unfold AE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -split. -unfold weak_eq. change_rewrite. -left. split; auto. -unfold same_type. left. split. exists N0. auto. exists N0. auto. -unfold s' in H1. -generalize (VertexSet.remove_3 H1). intro H2. -exists (snd_ext e', snd_ext e, Some N0). -split. -left. unfold AE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -apply (sym_pmap g). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -unfold same_type. left. split; exists N0; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -unfold same_type. left. split; exists N0; auto. -elim n1. intuition. intuition. -rewrite MapFacts.add_neq_o in H0. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH s'). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H0. -case_eq (VertexMap.find (fst_ext e') m); intros. rewrite H1 in H0. -exists (fst_ext e', snd_ext e', Some N0). -split. -left. apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -unfold adj_set. fold m. rewrite H1. assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split ; auto. -left. split; exists N0; auto. -rewrite H1 in H0. elim (VertexSet.empty_1 H0). -simpl in H0. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 (fst_ext e')). clear H1. intro H1. inversion H1; clear H1. -rewrite <-H3 in *. elim (VertexSet.empty_1 H0). -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H3. inversion H3; subst; clear H3. -rewrite <-H2 in H0. -rewrite H4 in H0. clear H4. -destruct (proj1 (Props.Dec.F.add_iff _ _ _ ) H0). -exists (fst_ext e', snd_ext e, Some N0). -split. -left. apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -apply (sym_pmap g). -assert (VertexSet.In (fst_ext e') s'). -apply HH. left. auto. -unfold s' in H3. apply (VertexSet.remove_3 H3). -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -unfold same_type. left. split; exists N0; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -left. split; exists N0; auto. -elim n2. intuition. -exists (fst_ext e', snd_ext e', Some N0). -split. -left. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj1 (In_graph_edge_in_ext _ _ Hin)). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -left. split; exists N0; auto. -intuition. -auto. - -rewrite MapFacts.add_neq_o in H3. rewrite <-H3 in IHl. -apply IHl. rewrite <-H2 in H0. rewrite H4 in H0. auto. -intros. apply HH. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -auto. -auto. - -generalize (IE_weights _ _ H0). intro Hw. -unfold IE in H0. simpl in H0. -rewrite (edge_eq e') in H0. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map _ _ q p)) H0). -clear H0. intro H0. -unfold imap_merge in H0. unfold map_merge in H0. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (imap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (imap g))))) in *. -set (m := imap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set in H0. -destruct (Vertex.eq_dec (fst_ext e') (snd_ext e)). -rewrite MapFacts.remove_eq_o in H0. -elim (VertexSet.empty_1 H0). intuition. -rewrite MapFacts.remove_neq_o in H0. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o in H0. -unfold s in H0. -destruct (VertexSet.union_1 H0). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -exists (fst_ext e', snd_ext e', None). split. -right. unfold IE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. left; split; auto. -right; split; unfold interf_edge; simpl; auto. -unfold s' in H1. -generalize (VertexSet.remove_3 H1). intro H2. -exists (snd_ext e', snd_ext e, None). -split. -right. unfold IE. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -apply (sym_imap g). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge;split; simpl; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -right. unfold interf_edge; split; simpl; auto. -elim n1. intuition. -intuition. -rewrite MapFacts.add_neq_o in H0. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH s'). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H0. -case_eq (VertexMap.find (fst_ext e') m); intros. rewrite H1 in H0. -exists (fst_ext e', snd_ext e', None). -split. -right. apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -unfold adj_set. fold m. rewrite H1. assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (n r). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge; split; simpl; auto. -rewrite H1 in H0. elim (VertexSet.empty_1 H0). -simpl in H0. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H1 (fst_ext e')). clear H1. intro H1. inversion H1; clear H1. -rewrite <-H3 in *. elim (VertexSet.empty_1 H0). -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H3. inversion H3; subst; clear H3. -rewrite <-H2 in H0. -rewrite H4 in H0. clear H4. -destruct (proj1 (Props.Dec.F.add_iff _ _ _ ) H0). -exists (fst_ext e', snd_ext e, None). -split. -right. apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -apply (sym_imap g). -assert (VertexSet.In (fst_ext e') s'). -apply HH. left. auto. -unfold s' in H3. apply (VertexSet.remove_3 H3). -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -right. split; auto. -right. unfold interf_edge; split; simpl; auto. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge; split; simpl; auto. -elim n2. intuition. -exists (fst_ext e', snd_ext e', None). -split. -right. apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -generalize (VertexSet.remove_3 H1). clear H1. intro H1. -unfold adj_set. rewrite (MapFacts.find_o _ e0). assumption. -unfold redirect. change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj1 (In_graph_edge_in_ext _ _ Hin)). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). -elim (merge_1 e g p q). -rewrite <-r. apply (proj2 (In_graph_edge_in_ext _ _ Hin)). -unfold weak_eq. change_rewrite. -split. -left. split; auto. -right. unfold interf_edge; split; simpl; auto. -intuition. - -rewrite MapFacts.add_neq_o in H3. rewrite <-H3 in IHl. -apply IHl. rewrite <-H2 in H0. rewrite H4 in H0. auto. -intros. apply HH. auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. - -auto. -auto. -Qed. - -Lemma resolve_conflicts_map_5 : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -VertexSet.In x (adj_set y (map_merge e (pmap g))) -> -~VertexSet.In x (adj_set y (imap_merge e g)) -> -VertexSet.In x (adj_set y (resolve_conflicts (fst_ext e) (map_merge e (pmap g)) - (adj_set (fst_ext e) (map_merge e (pmap g))) - (adj_set (fst_ext e) (imap_merge e g)))). - -Proof. -intros x y e g p q H H0. -unfold resolve_conflicts. -set (f := (fun (x0 : VertexSet.elt) (m : VertexMap.t VertexSet.t) => - VertexMap.add x0 - (VertexSet.remove (fst_ext e) - (adj_set x0 (map_merge e (pmap g)))) m)) in *. -set (m := map_merge e (pmap g)) in *. -set (s' := (VertexSet.diff (adj_set (fst_ext e) m) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -set (s := (VertexSet.inter (adj_set (fst_ext e) m) - (adj_set (fst_ext e) (imap_merge e g)))) in *. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro. -generalize (H1 s y). clear H1. intro HH. -generalize VertexSet.elements_2. intro. -generalize (H1 s y). clear H1. intro HHH. -induction (VertexSet.elements s). simpl. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set. rewrite (MapFacts.find_o _ e0). -rewrite MapFacts.add_eq_o. -apply VertexSet.diff_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. -unfold adj_set. rewrite MapFacts.add_neq_o. assumption. -auto. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). -intro. generalize (H1 y). clear H1. intro H1. simpl. inversion H1; clear H1. -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H4. congruence. intuition. -destruct (Vertex.eq_dec y (fst_ext e)). unfold adj_set. -rewrite MapFacts.add_eq_o. -unfold adj_set in IHl. -rewrite MapFacts.add_eq_o in IHl. -apply IHl. intro. -generalize (HH H1). intro. -inversion H2; subst. -elim (n H6). -auto. -intro. apply HHH. right. auto. -intuition. -intuition. - -rewrite MapFacts.add_neq_o in H4. -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. -rewrite <-H4 in IHl. -assert (VertexSet.In x VertexSet.empty). -apply IHl. intro. -generalize (HH H1). intro. -inversion H2; subst. -elim (n H6). -auto. -intro. apply HHH. right. auto. -elim (VertexSet.empty_1 H1). -auto. -auto. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H3. unfold f in H3. -destruct (Vertex.eq_dec y a). -rewrite MapFacts.add_eq_o in H3. -unfold adj_set. destruct (Vertex.eq_dec y (fst_ext e)). -rewrite MapFacts.add_eq_o. -apply VertexSet.diff_3. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). assumption. -intuition. - -rewrite MapFacts.add_neq_o. rewrite <-H2. rewrite H4. -inversion H3; subst; clear H3. -apply VertexSet.remove_2. - -intro H5. -assert (VertexSet.In y s). -apply HHH. left. auto. -generalize (VertexSet.inter_2 H1). intro. -unfold adj_set in H3. rewrite (MapFacts.find_o _ H5) in H3. -generalize (sym_imap_merge_map e g p q _ _ H3). intro. -elim H0. assumption. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -auto. -auto. -intuition. - -rewrite MapFacts.add_neq_o in H3. -destruct (Vertex.eq_dec y (fst_ext e)). -unfold adj_set in IHl. -rewrite MapFacts.add_eq_o in IHl. -unfold adj_set. rewrite MapFacts.add_eq_o. -apply IHl. - -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -auto. - -intro. apply HHH. right. auto. -intuition. -intuition. - -unfold adj_set in IHl. rewrite MapFacts.add_neq_o in IHl. -unfold adj_set. rewrite MapFacts.add_neq_o. -rewrite <-H2. -rewrite <-H3 in IHl. -rewrite H4. -apply IHl. - -intro. generalize (HH H1). intro. -inversion H5; subst. -elim (n H7). -auto. - -intro. apply HHH. right. auto. -auto. -auto. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x0 z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x0 y0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x0 s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x0 a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H1. auto. auto. -Qed. - -Lemma merge_3_aux : forall e e' g, -aff_edge e -> -In_graph_edge e g -> -In_graph_edge e' g -> -aff_edge e' -> -~Edge.eq e e' -> -VertexSet.In (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (adj_set (fst_ext (redirect (snd_ext e) (fst_ext e) e')) (map_merge e (pmap g))). - -Proof. -intros e e' g q p H0 H1 Hdiff. generalize I. intro H. -assert (get_weight e = Some N0) as Hw. -destruct p. apply (AE_weights _ _ H2). -generalize (IE_weights _ _ H2). inversion q. congruence. -assert (get_weight e' = Some N0) as Hw'. -destruct H0. apply (AE_weights _ _ H0). -generalize (IE_weights _ _ H0). inversion H1. congruence. -assert (EdgeSet.In (fst_ext e, snd_ext e, Some N0) (AE g)) as He. -destruct p. rewrite (edge_eq e) in H2. -generalize (AE_weights _ _ H2). intro. simpl in H3. -rewrite H3 in H2. assumption. -generalize (IE_weights _ _ H2). inversion q. congruence. -assert (VertexSet.In (snd_ext e) (adj_set (fst_ext e) (pmap g))) as Hee. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assumption. -assert (EdgeSet.In (fst_ext e', snd_ext e', Some N0) (AE g)) as He'. -destruct H0. rewrite (edge_eq e') in H0. -generalize (AE_weights _ _ H0). intro. simpl in H2. -rewrite H2 in H0. assumption. -generalize (IE_weights _ _ H0). intro H2. inversion H1. congruence. -assert (VertexSet.In (snd_ext e') (adj_set (fst_ext e') (pmap g))) as Hee'. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assumption. -assert (forall a b, VertexSet.In a (adj_set b (map_merge e (pmap g))) -> - VertexSet.In b (adj_set a (map_merge e (pmap g)))) as Hsym. -intros; apply sym_map_merge_pmap; auto. - -unfold redirect. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_3. -unfold s'. -apply VertexSet.remove_2. -intro. change_rewrite. -elim Hdiff. -rewrite (edge_eq e). rewrite edge_comm. apply eq_ordered_eq. -rewrite (edge_eq e'). unfold E.eq; change_rewrite; simpl; intuition. -rewrite Hw. rewrite Hw'. apply OptionN_as_OT.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -fold (adj_set (fst_ext e') m). -assumption. change_rewrite. intuition. -change_rewrite. -apply (In_graph_edge_diff_ext _ _ p). - -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)); change_rewrite. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -induction (VertexSet.elements s'). simpl. -assert (InA Vertex.eq (fst_ext e') nil). -apply HH. -unfold s'. apply VertexSet.remove_2. -intro. elim Hdiff. -apply eq_ordered_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). -unfold E.eq; change_rewrite; simpl; intuition. -rewrite Hw. rewrite Hw'. apply OptionN_as_OT.eq_refl. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym r)). -apply (sym_pmap g). assumption. -inversion H2. - -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. -intro. generalize (HH H2). clear HH. intro HH. -inversion HH; subst. -elim (n0 H6). -assumption. -auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_1. intuition. intuition. -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -intro. generalize (HH H2). clear HH. intro HH. inversion HH; subst. -elim (n0 H7). -assumption. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -intro. elim Hdiff. -apply eq_ordered_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). -unfold E.eq; change_rewrite; simpl; intuition. -rewrite Hw. rewrite Hw'. apply OptionN_as_OT.eq_refl. - -intuition. -unfold map_merge. -set (s := (VertexSet.union - (VertexSet.remove (snd_ext e) (adj_set (fst_ext e) (pmap g))) - (VertexSet.remove (fst_ext e) (adj_set (snd_ext e) (pmap g))))) in *. -set (m := pmap g) in *. -set (f := (fun (y : VertexSet.elt) (m' : VertexMap.t VertexSet.t) => - VertexMap.add y - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (adj_set y m))) m')) in *. -set (s' := VertexSet.remove (fst_ext e) (adj_set (snd_ext e) m)) in *. -unfold adj_set. -rewrite MapFacts.remove_neq_o. -destruct (Vertex.eq_dec (fst_ext e') (fst_ext e)). -rewrite MapFacts.add_eq_o. -unfold s. -apply VertexSet.union_2. -apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -assumption. -intuition. - -rewrite MapFacts.add_neq_o. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a). -(* -generalize VertexSet.elements_1. intro HH. -generalize (HH s' (fst_ext e')). clear HH. intro HH. -*) -induction (VertexSet.elements s'). simpl. assumption. -cut (EqualSetMap (fold_left f' (a :: l) m) (f' (fold_left f' l m) a)). intro. -generalize (H2 (fst_ext e')). clear H2. intro H2. simpl. inversion H2;clear H2. -set (tmp := fold_left f' l m) in *. -unfold f' in H5. unfold f in H5. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H5. congruence. intuition. -rewrite MapFacts.add_neq_o in H5. rewrite <-H5 in IHl. -apply IHl. auto. - -set (tmp := fold_left f' l m) in *. -unfold f' in H4. unfold f in H4. -destruct (Vertex.eq_dec (fst_ext e') a). -rewrite MapFacts.add_eq_o in H4. -rewrite H5. inversion H4; subst; clear H4. -apply VertexSet.add_2. apply VertexSet.remove_2. intuition. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). assumption. -intuition. - -rewrite MapFacts.add_neq_o in H4. -rewrite <-H4 in IHl. -rewrite H5. apply IHl. -auto. - -apply fold_left_assoc_map. -unfold EqualSetMap. unfold f'. unfold f. -intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_sym e1)). -rewrite (MapFacts.find_o _ (Vertex.eq_sym e0)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor; apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -unfold f'. unfold f. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H2. auto. auto. - -auto. -intuition. -Qed. - -Lemma merge_5 : forall e e' g p q, -In_graph_edge e' g -> -~Edge.eq e e' -> -aff_edge e' -> -~Interfere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q) -> -Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q). - -Proof. -intros e e' g p q H0 H1 H2 H3. generalize I. intro H. -unfold Prefere. exists N0. left. -unfold merge. unfold AE. simpl. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -apply resolve_conflicts_map_5. -assumption. -assumption. -apply merge_3_aux; assumption. -intro. elim H3. unfold Interfere. right. unfold merge. -unfold IE. simpl. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -assumption. -Qed. -(* -unfold change. -destruct (OTFacts.eq_dec (fst_ext e') (snd_ext e)). change_rewrite. -apply VertexSet.remove_2. -apply (In_graph_edge_diff_ext e g p). -apply (proj1 (In_graph_edge_in_ext _ _ p)). -destruct (OTFacts.eq_dec (snd_ext e') (snd_ext e)). change_rewrite. -apply VertexSet.remove_2. -intro. elim (In_graph_edge_diff_ext e' g H0). -apply Vertex.eq_trans with (y := snd_ext e); auto. -apply (proj1 (In_graph_edge_in_ext _ _ H0)). -apply VertexSet.remove_2. -auto. -apply (proj1 (In_graph_edge_in_ext _ _ H0)). -Qed. -*) - -Lemma precolored_merge2 : forall x e g p q, -(VertexSet.In x (VertexSet.remove (snd_ext e) (precolored g)) <-> - VertexSet.In x (precolored (merge e g p q))). - -Proof. -intros x e g HH q. split; intros. -unfold merge. -unfold precolored. simpl. unfold is_precolored. simpl. -apply VertexSet.filter_3. -unfold compat_bool;intros. -rewrite (mreg_ext _ _ H0). auto. -apply VertexSet.remove_2. -intro H1. elim (VertexSet.remove_1 H1 H). -apply (VertexSet.filter_1 (compat_bool_is_precolored _) (VertexSet.remove_3 H)). -generalize (VertexSet.filter_2 (compat_bool_is_precolored _) (VertexSet.remove_3 H)). intro H0. -assumption. - -unfold precolored, is_precolored in *. unfold merge in H. simpl in H. -apply VertexSet.remove_2. intro. -generalize (VertexSet.filter_1 (compat_bool_is_precolored (merge e g HH q)) H). intro. -elim (VertexSet.remove_1 H0 H1). -apply VertexSet.filter_3. -unfold compat_bool. apply mreg_ext. -generalize (VertexSet.filter_1 (compat_bool_is_precolored (merge e g HH q)) H). intros. -apply (VertexSet.remove_3 H0). -apply (VertexSet.filter_2 (compat_bool_is_precolored (merge e g HH q)) H). -Qed. - -(* spec of delete_preference_edges *) -Lemma delete_preference_edges_prec : forall r g Hin, -VertexSet.Equal (precolored (delete_preference_edges r g Hin)) (precolored g). - -Proof. -intros. -unfold delete_preference_edges. -unfold precolored. simpl. -unfold is_precolored. simpl. -apply VertexSet.eq_refl. -Qed. - -Lemma delete_preference_edges_1 : forall e r g Hin, -In_graph_edge e (delete_preference_edges r g Hin) -> In_graph_edge e g. - -Proof. -intros. -unfold delete_preference_edges in H. -destruct H;[left; generalize (AE_weights _ _ H); unfold AE in *| - right; generalize (IE_weights _ _ H); unfold IE in *]; simpl in H; intros. -rewrite (edge_eq e) in H. -rewrite H0 in H. generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap_delete_preferences r g)) H). intro. -rewrite (edge_eq e). -rewrite H0. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g))). -apply (delete_preference_sub _ _ _ _ H1). -assumption. -Qed. - -Lemma IE_delete_preference_eq : forall x g H, -EdgeSet.Equal (IE g) (IE (delete_preference_edges x g H)). - -Proof. -intros. -unfold delete_preference_edges. -destruct (In_graph_dec x g). -unfold IE. simpl. apply EdgeSet.eq_refl. -apply EdgeSet.eq_refl. -Qed. - -Lemma delete_preference_edges_2 : forall e r g Hin, -In_graph_edge e g -> -~incident e r -> -In_graph_edge e (delete_preference_edges r g Hin). - -Proof. -intros. -destruct H;[left|right]. -generalize (AE_weights _ _ H). intro Hw. -unfold AE in *. -rewrite (edge_eq e). rewrite Hw. -unfold delete_preference_edges. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap_delete_preferences r g))). -unfold pmap_delete_preferences. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set r (pmap g))). simpl. -unfold adj_set. rewrite InterfFacts.add_neq_o. -rewrite (edge_eq e) in H. rewrite Hw in H. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g)) H). -intro H1. elim H0. left. auto. - -cut (EqualSetMap - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e (pmap g))) a) (a :: l) - (VertexMap.add r VertexSet.empty (pmap g))) - (VertexMap.add a - (VertexSet.remove r - (adj_set a (pmap g))) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e (pmap g))) a) l - (VertexMap.add r VertexSet.empty (pmap g))))). intro. -unfold EqualSetMap in H1. generalize (H1 (fst_ext e)). clear H1. intro H1. -inversion H1. simpl. unfold adj_set in *. rewrite <-H3. -destruct (Vertex.eq_dec (fst_ext e) a). -rewrite InterfFacts.add_eq_o in H4. inversion H4. -intuition. -rewrite InterfFacts.add_neq_o in H4. rewrite <-H4 in *. -elim (VertexSet.empty_1 IHl). -auto. -simpl. unfold adj_set in *. rewrite <-H2. -rewrite H4. clear H1. clear H2. clear H4. -destruct (Vertex.eq_dec (fst_ext e) a). -rewrite InterfFacts.add_eq_o in H3. -inversion H3. clear H2. -apply VertexSet.remove_2. -intro. elim H0. right. auto. -clear H3. -rewrite (edge_eq e) in H. rewrite Hw in H. -rewrite (InterfFacts.find_o _ (Vertex.eq_sym e0)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). -intuition. -rewrite (InterfFacts.add_neq_o) in H3. -case_eq (VertexMap.find (elt:=VertexSet.t) (fst_ext e) - (fold_left - (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e - (VertexSet.remove r - match VertexMap.find (elt:=VertexSet.t) e (pmap g) with - | Some x => x - | None => VertexSet.empty - end) a) l (VertexMap.add r VertexSet.empty (pmap g)))); -intros; rewrite H1 in *. -inversion H3. assumption. -inversion H3. -auto. - -apply fold_left_assoc_map. -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x z). -rewrite InterfFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite InterfFacts.add_eq_o. -constructor. -generalize (Vertex.eq_trans (Vertex.eq_sym e0) e1). intro HHH. -unfold adj_set. -rewrite (InterfFacts.find_o _ HHH). apply VertexSet.eq_refl. -intuition. - -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -auto. -intuition. - -rewrite InterfFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. - -intros. -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x y). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply EqualSetMap_refl. -auto. -auto. -auto. -auto. - -unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x a0). -rewrite InterfFacts.add_eq_o. -rewrite InterfFacts.add_eq_o. -constructor. apply VertexSet.eq_refl. -intuition. -intuition. -rewrite InterfFacts.add_neq_o. -rewrite InterfFacts.add_neq_o. -apply (H1 x). -auto. -auto. - -rewrite <-IE_delete_preference_eq. auto. -Qed. - -Lemma V_delete_preference_eq : forall x g Hin, -VertexSet.Equal (V g) (V (delete_preference_edges x g Hin)). - -Proof. -intros. -unfold delete_preference_edges. -destruct (In_graph_dec x g); simpl; apply VertexSet.eq_refl. -Qed. - -Lemma in_delete_preference_not_incident : forall e r g Hdep, -EdgeSet.In e (AE (delete_preference_edges r g Hdep)) -> ~incident e r. - -Proof. -intros. -generalize (AE_weights _ _ H). intro Hw. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H). intro Hin. -unfold AE in H. -rewrite (edge_eq e) in H. rewrite Hw in H. -unfold delete_preference_edges in H. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap_delete_preferences r g)) H). clear H. intro H. -unfold pmap_delete_preferences in H. -rewrite VertexSet.fold_1 in H. -generalize VertexSet.elements_1. intro HH. -generalize (HH (adj_set r (pmap g)) (fst_ext e)). clear HH. intro HH. -generalize VertexSet.elements_2. intro HH1. -generalize (HH1 (adj_set r (pmap g))). clear HH1. intro HH1. - -assert (forall z, In z (VertexSet.elements (adj_set r (pmap g))) -> ~Vertex.eq r z) as Hneq. -clear H HH Hin Hw Hdep. -intros. intro H0. -induction (VertexSet.elements (adj_set r (pmap g))). inversion H. -simpl in H. destruct H. subst. -assert (VertexSet.In r (adj_set r (pmap g))). -apply HH1. left. auto. -elim (not_eq_extremities g r r). -right. assumption. -auto. -apply IHl. -intros. apply HH1. auto. -assumption. - -induction (VertexSet.elements (adj_set r (pmap g))). simpl in H. -unfold adj_set in H. -destruct (Vertex.eq_dec (fst_ext e) r). -rewrite InterfFacts.add_eq_o in H. -elim (VertexSet.empty_1 H). intuition. -rewrite InterfFacts.add_neq_o in H. -destruct (Vertex.eq_dec (snd_ext e) r). -assert (InA Vertex.eq (fst_ext e) nil). -apply HH. -apply (proj1 (edgemap_to_edgeset_charac (pmap g) r (fst_ext e) (Some N0) (sym_pmap g))). -assert (eq (r,fst_ext e, Some N0) e). -rewrite edge_comm. -apply eq_ordered_eq. -unfold E.eq; simpl; intuition. apply Regs.eq_refl. -rewrite <-Hw. apply OptionN_as_OT.eq_refl. -rewrite H0. -destruct Hin. -generalize (delete_preference_edges_1 e r g Hdep H2). clear H2. intro H2. -destruct H2. -unfold AE in H2. assumption. -generalize (IE_weights _ _ H2). intro Hw'. rewrite Hw' in Hw. inversion Hw. -inversion H0. -intro H0. destruct H0. -elim n. auto. -elim n0. auto. -auto. - -set (f := (fun (a : VertexMap.t VertexSet.t) (e : VertexSet.elt) => - VertexMap.add e (VertexSet.remove r (adj_set e (pmap g))) a)) in *. -set (s := VertexMap.add r VertexSet.empty (pmap g)) in *. - -assert (EqualSetMap (fold_left f (a :: l) s) (f (fold_left f l s) a)). -apply fold_left_assoc_map. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x z). -rewrite MapFacts.add_eq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -constructor. -apply Props.Equal_remove. -unfold adj_set. -rewrite (MapFacts.find_o _ (Vertex.eq_trans (Vertex.eq_sym e0) e1)). -apply VertexSet.eq_refl. intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -auto. -intuition. -rewrite MapFacts.add_neq_o. -destruct (Vertex.eq_dec x y). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -destruct (VertexMap.find x s0); constructor. -apply VertexSet.eq_refl. -auto. -auto. -auto. -auto. - -unfold f. unfold EqualSetMap. intros. -destruct (Vertex.eq_dec x a0). -rewrite MapFacts.add_eq_o. -rewrite MapFacts.add_eq_o. -constructor. -apply VertexSet.eq_refl. -intuition. -intuition. -rewrite MapFacts.add_neq_o. -rewrite MapFacts.add_neq_o. -apply H0. -auto. -auto. - -generalize (H0 (fst_ext e)). clear H0. intro H0. -simpl in H. unfold adj_set in H. -inversion H0. subst. -rewrite <-H2 in H. -elim (VertexSet.empty_1 H). -rewrite <-H1 in H. -set (tmp := fold_left f l s) in *. -unfold f in H2. -destruct (Vertex.eq_dec (fst_ext e) a). -rewrite MapFacts.add_eq_o in H2. -intro. -inversion H2. subst. clear H2. -rewrite H3 in H. -destruct H4. - -elim (Hneq a). -left. auto. -apply (Vertex.eq_trans H2 e0). -elim (VertexSet.remove_1 H2 H). -intuition. - -rewrite MapFacts.add_neq_o in H2. -inversion H2. subst. clear H2. -apply IHl. -clear IHl. -unfold adj_set. rewrite <-H5. -rewrite <-H3. assumption. -intro. -assert (InA Vertex.eq (fst_ext e) (a :: l)). -apply HH. assumption. -inversion H4; subst. -elim (n H7). -assumption. -intros. apply HH1. auto. -intros. apply Hneq. right. auto. -auto. -Qed. - -Add Morphism In_graph_edge : In_graph_edge_m. - -Proof. -unfold In_graph_edge;intros x y H g. -fold (eq x y) in H. -split;intro H0;[rewrite <-H|rewrite H];assumption. -Qed. - -(* There cannot exist both an interference and - a preference between two vertices *) -Lemma interf_pref_conflict : forall x y g, -Prefere x y g /\ Interfere x y g -> False. - -Proof. -intros. -unfold Prefere in H. unfold Interfere in H. -destruct H. -apply (simple_graph g x y). -split. -destruct H0. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H0). intro H1. -destruct H1 as [H1 _]. -unfold aff_edge in H1. -destruct H1. inversion H1. -unfold IE in H0. -rewrite edge_comm in H0. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g)) H0). -destruct H. destruct H. generalize (AE_weights _ _ H). simpl. intro. -rewrite H1 in H. -rewrite edge_comm in H. unfold AE in H. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H). intro H1. -destruct H1 as [H1 _]. -unfold interf_edge in H1. inversion H1. -Qed. - -Lemma is_simple_graph : forall e e' g, -In_graph_edge e g -> -In_graph_edge e' g -> -weak_eq e e' -> -eq e e'. - -Proof. -intros. -unfold weak_eq in H1. -destruct H; destruct H0. -generalize (AE_weights _ _ H); intros. -generalize (AE_weights _ _ H0); intros. -destruct H1; destruct H1. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_comm_eq. -elim (interf_pref_conflict (fst_ext e') (snd_ext e') g). -unfold Prefere, Interfere; split. -destruct H1; destruct H1. exists N0. -rewrite <-(AE_weights _ _ H). -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_aff_edge_in_AE in H. destruct H. intuition. -exists N0. rewrite <-(AE_weights _ _ H). -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_comm_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_aff_edge_in_AE in H. destruct H. intuition. -rewrite <-(IE_weights _ _ H0). rewrite In_graph_interf_edge_in_IE in H0. destruct H0. rewrite (edge_eq e') in H2. intuition. -elim (interf_pref_conflict (fst_ext e') (snd_ext e') g). -unfold Prefere, Interfere; split. -exists N0. rewrite <-(AE_weights _ _ H0). -rewrite (edge_eq e') in H0. rewrite In_graph_aff_edge_in_AE in H0. destruct H0. auto. -rewrite <-(IE_weights _ _ H). -destruct H1; destruct H1. -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_interf_edge_in_IE in H. destruct H. auto. -assert (eq (fst_ext e', snd_ext e', get_weight e) (fst_ext e, snd_ext e, get_weight e)) by Eq_comm_eq. -rewrite H3. rewrite <-(edge_eq e). rewrite In_graph_interf_edge_in_IE in H. destruct H. auto. -generalize (IE_weights _ _ H); intros. -generalize (IE_weights _ _ H0); intros. -destruct H1; destruct H1. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_eq. -rewrite (edge_eq e). rewrite (edge_eq e'). rewrite H2. rewrite H3. Eq_comm_eq. -Qed. - -Lemma is_precolored_ext : forall x y g, -Vertex.eq x y -> -is_precolored x g = is_precolored y g. - -Proof. -intros. apply compat_bool_is_precolored. assumption. -Qed. - -(* This module respects the interface *) - -(* A vertex x is in (remove_vertex r g) iff it is in g - and it is different from r*) -Lemma In_remove_vertex : forall x r g, -In_graph x (remove_vertex r g) <-> (In_graph x g /\ ~Vertex.eq x r). - -Proof. -split; intros. -split. apply in_remove_in with (r:=r). auto. -intro. rewrite H0 in H. elim (not_in_remove r g H). -destruct H. apply In_remove; auto. -Qed. - -(* The precolored vertices of (remove_vertex r g) are - the precolored vertices of g, minus r *) -Lemma precolored_remove_vertex : forall r g, -VertexSet.Equal (precolored (remove_vertex r g)) - (VertexSet.remove r (precolored g)). - -Proof. -split; intros. -rewrite precolored_remove_vertex2; auto. -rewrite <-precolored_remove_vertex2; auto. -Qed. - -(* An edge e is in (remove_vertex r g) iff it is in g - and is not incident to r *) -Lemma In_remove_edge : forall e r g, -In_graph_edge e (remove_vertex r g) <-> (In_graph_edge e g /\ ~incident e r). - -Proof. -split; intros. -split. apply in_remove_in_edge with (r:=r). auto. -intro. elim (not_in_remove_edge _ _ _ H0 H). -destruct H. apply In_remove_edge_; auto. -Qed. - -(* Specification of merge *) - -(* A vertex is in (merge e g p q) iff x is in g and - x is not equal to the second endpoint of g *) -Lemma In_merge_vertex : forall g e x p q, -In_graph x (merge e g p q) <-> (In_graph x g /\ ~Vertex.eq x (snd_ext e)). - -Proof. -split; intros. -split. apply (In_merge_in g e x p q H). -intro. rewrite H0 in H. elim (merge_1 e g p q H). -destruct H. apply merge_2; auto. -Qed. - -(* If an interference edge e' is in the graph g then - its redirection from the second endpoint of e - to the first endpoint of e is in (merge e g p q) *) -Lemma In_merge_interf_edge : forall e e' g p q, -In_graph_edge e' g -> -interf_edge e' -> -In_graph_edge (redirect (snd_ext e) (fst_ext e) e') (merge e g p q). - -Proof. -intros. apply merge_3; auto. -Qed. - -(* If a preference edge e' different from e is in the graph g, - and iff the endpoints of its redirection from the second - endpoint of e to the first endpoint of e do not interfere, - then these endpoints are linked with an affinity edge, whose - weight may be different from the one of e' *) -Lemma In_merge_pref_edge : forall e e' g p q, -In_graph_edge e' g -> -~Edge.eq e e' -> -aff_edge e' -> -~Interfere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q) -> - Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) e')) - (snd_ext (redirect (snd_ext e) (fst_ext e) e')) - (merge e g p q). - -Proof. -intros. apply merge_5; auto. -Qed. - -(* Inversely, if e' is an edge of (merge e g p q) then there exists - an edge a of g, such that e' is weakly equal to the redirection - of a from the second endpoint of e to the first endpoint of e *) -Lemma In_merge_edge_inv : forall e e' g p q, -In_graph_edge e' (merge e g p q) -> -exists a, In_graph_edge a g /\ weak_eq e' (redirect (snd_ext e) (fst_ext e) a) /\ - same_type a e'. - -Proof. -intros. apply (merge_4 e e' g p q); auto. -Qed. - -(* The precolored vertices of (merge e g p q) are the ones of g, - minus the second endpoint of e *) -Lemma precolored_merge : forall e g p q, -VertexSet.Equal (precolored (merge e g p q)) - (VertexSet.remove (snd_ext e) (precolored g)). - -Proof. -split; intros. -rewrite <-precolored_merge2 in H. auto. -rewrite <-precolored_merge2. auto. -Qed. - -(* Specification of delete_preference_edges *) - -(* A vertex is in (delete_preference_edges r g p) iff it is in g *) -Lemma In_delete_preference_edges_vertex : forall x r g p, -In_graph x (delete_preference_edges r g p) <-> In_graph x g. - -Proof. -unfold In_graph. split; intros. -rewrite <-V_delete_preference_eq in H. auto. -rewrite <-V_delete_preference_eq. auto. -Qed. - -(* The precolored vertices of (delete_preference_edges r g p) - iff it is precolored in g *) -Lemma precolored_delete_preference_edges : forall r g p, -VertexSet.Equal (precolored (delete_preference_edges r g p)) - (precolored g). - -Proof. -intros. apply delete_preference_edges_prec. -Qed. - -(* An edge e is in (delete_preference_edges r g p) iff - if is in g and it is not an affinity edge incident to r *) -Lemma In_delete_preference_edges_edge : forall e r g p, -In_graph_edge e (delete_preference_edges r g p) <-> -(In_graph_edge e g /\ ~(aff_edge e /\ incident e r)). - -Proof. -split; intros. -split. apply (delete_preference_edges_1 _ _ _ _ H). -intro. destruct H0. -assert (~incident e r). -apply (in_delete_preference_not_incident e r g p). -rewrite In_graph_aff_edge_in_AE. split; auto. -elim (H2 H1). -destruct H. destruct H. -rewrite In_graph_aff_edge_in_AE in H. destruct H. -apply delete_preference_edges_2; auto. -right. rewrite <-IE_delete_preference_eq. auto. -Qed. - -Definition interference_adj v g := -adj_set v (imap g). - -Definition preference_adj v g := -adj_set v (pmap g). - -(* Definition of the interference and preference degrees *) -Definition interf_degree g v := VertexSet.cardinal (interference_adj v g). -Definition pref_degree g v := VertexSet.cardinal (preference_adj v g). - -(* Definition of the low-degree function, - returns true iff the interference degree of v in g is strictly lower than K *) -Definition has_low_degree g K v := -if le_lt_dec K (interf_degree g v) then false else true. - -(* Definition of the move-related function, - returns true iff the vertex is move-related *) -Definition move_related g x := negb (VertexSet.is_empty (preference_adj x g)). - -Lemma in_pref_pref : forall x y g, -VertexSet.In x (preference_adj y g) -> -exists w, In_graph_edge (x,y,Some w) g. - -Proof. -intros. -unfold preference_adj in H. -exists N0. -left. unfold AE. -rewrite edge_comm. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_pmap g))). assumption. -Qed. - -Lemma pref_in_pref : forall x y g w, -In_graph_edge (x,y,Some w) g -> -VertexSet.In x (preference_adj y g). - -Proof. -intros. -unfold preference_adj. -destruct H. -generalize (AE_weights _ _ H). intro. -unfold AE in H. simpl in H0. rewrite H0 in H. -rewrite edge_comm in H. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H). -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H). intro. -destruct H0. -unfold interf_edge in H0. inversion H0. -Qed. - -Lemma in_interf_interf : forall x y g, -VertexSet.In x (interference_adj y g) -> -In_graph_edge (x,y,None) g. - -Proof. -intros. -unfold preference_adj in H. -right. unfold IE. -rewrite edge_comm. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _ (sym_imap g))). assumption. -Qed. - -Lemma interf_in_interf : forall x y g, -In_graph_edge (x,y,None) g -> -VertexSet.In x (interference_adj y g). - -Proof. -intros. -unfold interference_adj. -destruct H. -generalize (AE_weights _ _ H). intro. inversion H0. -rewrite edge_comm in H. apply (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_imap g)) H). -Qed. - -Lemma compat_interference_adj : forall x y g, -Vertex.eq x y -> -interference_adj x g = interference_adj y g. - -Proof. -intros. -unfold interference_adj. -unfold adj_set. -rewrite (InterfFacts.find_o _ H). reflexivity. -Qed. - -Lemma compat_preference_adj : forall x y g, -Vertex.eq x y -> -preference_adj x g = preference_adj y g. - -Proof. -intros. -unfold preference_adj. -unfold adj_set. -rewrite (InterfFacts.find_o _ H). reflexivity. -Qed. - -Lemma compat_bool_move : forall g, -compat_bool Vertex.eq (move_related g). - -Proof. -intros. -unfold compat_bool. -intros. -unfold move_related. -rewrite (compat_preference_adj _ _ _ H). -reflexivity. -Qed. - -(* characterisation of move relation *) - -Lemma move_related_charac : forall x g, -move_related g x = true -> -exists e, aff_edge e /\ In_graph_edge e g /\ incident e x. - -Proof. -intros. -unfold move_related in H. -case_eq (VertexSet.is_empty (preference_adj x g)); intros. -rewrite H0 in H. inversion H. generalize H0. clear H H0. intro H. -case_eq (VertexSet.choose (preference_adj x g)); intros. -exists (x,e,Some N0). -split. -exists N0. auto. -generalize (VertexSet.choose_1 H0). clear H0. intro H0. -split. -left. unfold AE. apply (proj2 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g))). -assumption. -left. auto. -generalize (VertexSet.choose_2 H0). clear H0. intro H0. -rewrite (VertexSet.is_empty_1 H0) in H. inversion H. -Qed. - -(* the inversion characterisation of move related *) - -Lemma move_related_charac2 : forall x g e, -aff_edge e -> -In_graph_edge e g -> -incident e x -> -move_related g x = true. - -Proof. -intros. -unfold move_related. -case_eq (VertexSet.is_empty (preference_adj x g)); intros. -generalize (VertexSet.is_empty_2 H2). clear H2. intro H2. -destruct H1. -elim (H2 (snd_ext e)). -apply pref_in_pref with (w:=N0). -rewrite edge_comm. -assert (eq (x,snd_ext e, Some N0) e). -apply eq_ordered_eq. -unfold E.eq; simpl; intuition. apply Regs.eq_refl. -destruct H0. generalize (AE_weights _ _ H0). intro. -rewrite <-H3. apply OptionN_as_OT.eq_refl. -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intros. -destruct H3. -unfold aff_edge in H. rewrite H3 in H. destruct H. inversion H. -rewrite H3. assumption. - -elim (H2 (fst_ext e)). -apply pref_in_pref with (w:=N0). -assert (eq (fst_ext e, x, Some N0) e). -apply eq_ordered_eq. -unfold E.eq; simpl; intuition. apply Regs.eq_refl. -destruct H0. generalize (AE_weights _ _ H0). intro. -rewrite <-H3. apply OptionN_as_OT.eq_refl. -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intros. -destruct H3. -unfold aff_edge in H. rewrite H3 in H. destruct H. inversion H. -rewrite H3. assumption. -auto. -Qed. -Definition WS := (VertexSet.t*VertexSet.t*VertexSet.t*EdgeSet.t)%type. - -Definition get_spillWL (w : WS) := fst (fst (fst w)). -Definition get_freezeWL (w : WS) := snd (fst (fst w)). -Definition get_simplifyWL (w : WS) := snd (fst w). -Definition get_movesWL (w : WS) := snd w. - -Lemma compat_bool_low : forall g palette, -compat_bool Vertex.eq (has_low_degree g palette). - -Proof. -unfold compat_bool. intros. -unfold has_low_degree, interf_degree. -rewrite (compat_interference_adj _ _ _ H). -reflexivity. -Qed. - -Definition WS_properties g K (WL : WS) : Prop := -(forall x, VertexSet.In x (get_spillWL WL) <-> has_low_degree g K x = false /\ In_graph x g /\ ~VertexSet.In x (precolored g)) /\ -(forall x, VertexSet.In x (get_freezeWL WL) <-> has_low_degree g K x = true /\ (move_related g) x = true /\ ~VertexSet.In x (precolored g)) /\ -(forall x, VertexSet.In x (get_simplifyWL WL) <-> has_low_degree g K x = true /\ (move_related g) x = false /\ In_graph x g /\ ~VertexSet.In x (precolored g)) /\ -(forall e, EdgeSet.In e (get_movesWL WL) <-> aff_edge e /\ In_graph_edge e g). - -Definition get_WL g palette := -let not_pre := VertexSet.diff (V g) (precolored g) in -let (low, spill) := VertexSet.partition (has_low_degree g palette) not_pre in -let (free, simp) := VertexSet.partition (move_related g) low in -(spill, free, simp, AE g). - -Module Import RegOTFacts := MyOTFacts Vertex. - -Lemma move_related_in : forall g x, -move_related g x = true -> -In_graph x g. - -Proof. -intros. -generalize (move_related_charac _ _ H). clear H. intro H. -destruct H as [e H]. destruct H. destruct H0. -apply (proj1 (extremities_pmap g x)). -destruct H0. -generalize (AE_weights _ _ H0). intro Hw. -unfold AE in *. -destruct H1. -rewrite (edge_eq e) in H0. -simpl in Hw. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H0). intro H2. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H2. -case_eq (VertexMap.find (fst_ext e) (pmap g)); intros; rewrite H3 in H2. -intro Helim. -rewrite (InterfFacts.find_o _ H1) in Helim. -rewrite H3 in Helim. inversion Helim. -elim (VertexSet.empty_1 H2). - -rewrite (edge_eq e) in H0. rewrite edge_comm in H0. -simpl in Hw. rewrite Hw in H0. -generalize (proj1 (edgemap_to_edgeset_charac _ _ _ _(sym_pmap g)) H0). intro H2. -apply (proj2 (InterfFacts.in_find_iff _ _)). -unfold adj_set in H2. -case_eq (VertexMap.find (snd_ext e) (pmap g)); intros; rewrite H3 in H2. -intro Helim. -rewrite (InterfFacts.find_o _ H1) in Helim. -rewrite H3 in Helim. inversion Helim. -elim (VertexSet.empty_1 H2). - -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intros. -destruct H2. unfold aff_edge in H. rewrite H2 in H. inversion H. inversion H4. -Qed. - -Lemma WS_prop_get : forall g palette, -WS_properties g palette (get_WL g palette). - -Proof. -intros. -unfold get_WL. -set (not_pre := VertexSet.diff (V g) (precolored g)) in *. -case_eq (VertexSet.partition (has_low_degree g palette) not_pre). -intros low spill H. -case_eq (VertexSet.partition (move_related g) low). -intros free simp H0. -unfold WS_properties. -unfold get_spillWL. unfold get_simplifyWL. unfold get_freezeWL. unfold get_movesWL. simpl. -assert (VertexSet.Equal low (VertexSet.filter (has_low_degree g palette) not_pre)). -assert (low = fst (VertexSet.partition (has_low_degree g palette) not_pre)). -rewrite H. auto. -rewrite H1. apply VertexSet.partition_1. apply compat_bool_low. -assert (VertexSet.Equal spill (VertexSet.filter (fun x => negb (has_low_degree g palette x)) not_pre)). -assert (spill = snd (VertexSet.partition (has_low_degree g palette) not_pre)). -rewrite H. auto. -rewrite H2. apply VertexSet.partition_2. apply compat_bool_low. -assert (VertexSet.Equal free (VertexSet.filter (move_related g) low)). -assert (free = fst (VertexSet.partition (move_related g) low)). -rewrite H0. auto. -rewrite H3. apply VertexSet.partition_1. apply compat_bool_move. -assert (VertexSet.Equal simp (VertexSet.filter (fun x => negb (move_related g x)) low)). -assert (simp = snd (VertexSet.partition (move_related g) low)). -rewrite H0. auto. -rewrite H4. apply VertexSet.partition_2. apply compat_bool_move. -split; intros. -split; intros. -rewrite H2 in H5. -split. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_low _ _)) H5). -destruct (has_low_degree g palette x); intuition. -split. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_low _ _)) H5). intro. -apply (VertexSet.diff_1 H6). -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_low _ _)) H5). intro. -apply (VertexSet.diff_2 H6). -rewrite H2. -apply VertexSet.filter_3. -apply compat_not_compat. apply compat_bool_low. -apply VertexSet.diff_3; intuition. -destruct H5. rewrite H5. auto. - -split; intros. -split; intros. -rewrite H3 in H5. -generalize (VertexSet.filter_1 (compat_bool_move _) H5). intro H6. -generalize (VertexSet.filter_2 (compat_bool_move _) H5). clear H5. intro H5. -rewrite H1 in H6. -generalize (VertexSet.filter_1 (compat_bool_low _ _) H6). intro H7. -generalize (VertexSet.filter_2 (compat_bool_low _ _) H6). clear H6. intro H6. -generalize (VertexSet.diff_2 H7). intuition. - -rewrite H3. apply VertexSet.filter_3. -apply compat_bool_move. -rewrite H1. -apply VertexSet.filter_3. -apply compat_bool_low. -apply VertexSet.diff_3. -apply move_related_in. intuition. -intuition. -intuition. -intuition. - -split;intros. -split;intros. -rewrite H4 in H5. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_move _)) H5). intro H6. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_move _)) H5). clear H5. intro H5. -rewrite H1 in H6. -generalize (VertexSet.filter_1 (compat_bool_low _ _) H6). intro H7. -generalize (VertexSet.filter_2 (compat_bool_low _ _) H6). clear H6. intro H6. -generalize (VertexSet.diff_2 H7). intuition. -destruct (move_related g x); intuition. -apply (VertexSet.diff_1 H7). - -rewrite H4. -apply VertexSet.filter_3. -apply compat_not_compat. apply compat_bool_move. -rewrite H1. apply VertexSet.filter_3. -apply compat_bool_low. -apply VertexSet.diff_3. -intuition. -intuition. -intuition. -destruct (move_related g x); intuition. -exact (In_graph_aff_edge_in_AE _ _). -Qed. - -Definition not_incident_edges x s g := -VertexSet.fold (fun y s' => EdgeSet.remove (x,y,Some N0) s') - (adj_set x (pmap g)) - s. - -Lemma not_incident_edges_1 : forall x e s g, -(forall y, EdgeSet.In y s -> aff_edge y /\ In_graph_edge y g) -> -(EdgeSet.In e (not_incident_edges x s g) <-> - EdgeSet.In e s /\ ~incident e x). - -Proof. -split; intros. -unfold not_incident_edges in H0. -rewrite VertexSet.fold_1 in H0. -assert (EdgeSet.In e s) as Hin. -induction (VertexSet.elements (adj_set x (pmap g))). simpl in H0. -assumption. -rewrite MEdgeFacts.fold_left_assoc in H0. -apply IHl. -apply (EdgeSet.remove_3 H0). - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -intros. apply RegRegProps.Equal_remove. assumption. - -split. -assumption. - -generalize VertexSet.elements_1. intro HH. -intro Helim. destruct Helim. -generalize (HH (adj_set x (pmap g)) (snd_ext e)). clear HH. intro HH. -induction (VertexSet.elements (adj_set x (pmap g))). simpl in H0. - -assert (VertexSet.In (snd_ext e) (adj_set x (pmap g))). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assert (eq e (x, snd_ext e, Some N0)). -assert (get_weight e = Some N0). - -generalize (H _ H0). intro. -destruct H2. unfold In_graph_edge in H3. -destruct H3. -apply (AE_weights g). assumption. -generalize (IE_weights g _ H3). intro. -destruct H2. congruence. -rewrite <-H2. Eq_eq. apply Regs.eq_refl. -rewrite <-H2. - -generalize (H _ H0). intro. -destruct H3. destruct H4. -assumption. -generalize (IE_weights g _ H4). intro. -destruct H3. congruence. - -generalize (HH H2). intro. inversion H3. - -rewrite MEdgeFacts.fold_left_assoc in H0. -apply IHl. -apply (EdgeSet.remove_3 H0). - -intro. generalize (HH H2). clear HH H2. intro H2. -inversion H2; subst. -assert (eq e (x, a, Some N0)). -rewrite (edge_eq e). -apply eq_ordered_eq. -constructor. simpl. split; intuition. -simpl. - -generalize (H _ Hin). intro H5. -destruct H5. -destruct H5. -rewrite (AE_weights g _ H5). apply OptionN_as_OT.eq_refl. -generalize (IE_weights g _ H5). intro. destruct H3. congruence. -elim (EdgeSet.remove_1 (eq_sym H3) H0). -assumption. - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -intros. apply RegRegProps.Equal_remove. assumption. - -generalize (HH (adj_set x (pmap g)) (fst_ext e)). clear HH. intro HH. -induction (VertexSet.elements (adj_set x (pmap g))). simpl in H0. - -assert (VertexSet.In (fst_ext e) (adj_set x (pmap g))). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assert (eq e (x, fst_ext e, Some N0)). -assert (get_weight e = Some N0). - -generalize (H _ H0). intro. -destruct H2. unfold In_graph_edge in H3. -destruct H3. -apply (AE_weights g). assumption. -generalize (IE_weights g _ H3). intro. -destruct H2. congruence. -rewrite <-H2. Eq_comm_eq. apply Regs.eq_refl. -rewrite <-H2. - -generalize (H _ H0). intro. -destruct H3. destruct H4. -assumption. -generalize (IE_weights g _ H4). intro. -destruct H3. congruence. - -generalize (HH H2). intro. inversion H3. - -rewrite MEdgeFacts.fold_left_assoc in H0. -apply IHl. -apply (EdgeSet.remove_3 H0). - -intro. generalize (HH H2). clear HH H2. intro H2. -inversion H2; subst. -assert (eq e (x, a, Some N0)). -rewrite (edge_eq e). -rewrite edge_comm. apply eq_ordered_eq. -constructor. simpl. split; intuition. -simpl. - -generalize (H _ Hin). intro H5. -destruct H5. -destruct H5. -rewrite (AE_weights g _ H5). apply OptionN_as_OT.eq_refl. -generalize (IE_weights g _ H5). intro. destruct H3. congruence. -elim (EdgeSet.remove_1 (eq_sym H3) H0). -assumption. - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H2). intro. -elim (EdgeSet.remove_1 H3 H4). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H3 H2). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H2)). -intros. apply RegRegProps.Equal_remove. assumption. - -unfold not_incident_edges. -rewrite VertexSet.fold_1. -induction (VertexSet.elements (adj_set x (pmap g))). simpl. -intuition. -rewrite MEdgeFacts.fold_left_assoc. -apply EdgeSet.remove_2. -destruct H0. -intro H2. elim H1. -destruct (eq_charac _ _ H2); destruct H3; change_rewrite. -left. auto. -right. auto. -assumption. - -intros. -split; intros. -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -apply EdgeSet.remove_2. intro. -generalize (EdgeSet.remove_3 H1). intro. -elim (EdgeSet.remove_1 H2 H3). -apply EdgeSet.remove_2. intro. -elim (EdgeSet.remove_1 H2 H1). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H1)). -intros. apply RegRegProps.Equal_remove. assumption. -Qed. - -Definition not_incident_merge x s es := -VertexSet.fold (fun y s' => EdgeSet.remove (x,y,Some N0) s') - es s. - -Lemma not_incident_merge_1 : forall x s es e, -EdgeSet.In e (not_incident_merge x es s) -> -EdgeSet.In e es. - -Proof. -unfold not_incident_merge; intros. -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (x, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements s). simpl in H. assumption. -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l es) in H. -unfold f' in H. unfold f in H. -apply (IHl (EdgeSet.remove_3 H)). - -unfold f'. unfold f. intros. -split; intros. -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H1. elim (EdgeSet.remove_1 H1 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). - -intros. -unfold f'. unfold f. -apply RegRegProps.Equal_remove. auto. -Qed. - - -Definition AE_merge_up newadj adj adjsnd e es := -let diff1 := VertexSet.diff adj newadj in -let diff2 := VertexSet.diff newadj adj in -let new_es := not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) es diff1) - adjsnd in -VertexSet.fold (fun y s'' => EdgeSet.add (fst_ext e,y,Some N0) s'') - diff2 new_es. - -Lemma not_incident_merge_2 : forall x y s s', -EdgeSet.In (x,y,Some N0) (not_incident_merge x s s') -> -~VertexSet.In y s'. - -Proof. -intros. -unfold not_incident_merge in H. -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (x, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1 in H. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH s' y). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H. -intro. generalize (HH H0). intro H1. inversion H1. -rewrite MEdgeFacts.fold_left_assoc in H. -set (tmp := fold_left f' l s) in *. -unfold f' in H. unfold f in H. -destruct (Vertex.eq_dec y a). -cut (eq (x,y,Some N0) (x,a, Some N0)). intro H0. -elim (EdgeSet.remove_1 (Edge.eq_sym H0) H). -apply eq_ordered_eq. constructor;simpl; try split; intuition. -apply OptionN_as_OT.eq_refl. -apply IHl. apply (EdgeSet.remove_3 H). -intro. generalize (HH H0). intro H1. -inversion H1; subst. -elim (n H3). -auto. - -unfold f'. unfold f. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H0)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H0). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H0)). - -unfold f'. unfold f. intros. -apply RegRegProps.Equal_remove. auto. -Qed. - -Lemma AE_merge_wl_1 : forall x y e g, -aff_edge e -> -In_graph_edge e g -> -(VertexSet.In x (adj_set y (pmap_merge e g (imap_merge e g))) <-> -(VertexSet.In x (adj_set y (map_merge e (pmap g))) /\ -~VertexSet.In x (adj_set y (imap_merge e g)))). - -Proof. -intros x y e g p q. split; intros. -split. -apply pmap_merge_sub. assumption. -intro. apply (simple_graph (merge e g q p) x y). intuition. -destruct H. -unfold merge. simpl. apply resolve_conflicts_map_5; auto. -Qed. - -Lemma not_in_diff_equiv : forall x s s', -~VertexSet.In x (VertexSet.diff s s') -> -VertexSet.In x s' \/ ~VertexSet.In x s. - -Proof. -intros. destruct (Props.In_dec x s). - destruct (Props.In_dec x s'). -left. auto. -elim H. apply VertexSet.diff_3; auto. -right. auto. -Qed. - -Lemma AE_aux_2 : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(forall b, EdgeSet.In b (AE_merge_up (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g) - (preference_adj (snd_ext e) g) - e s) - <-> - (~incident b (snd_ext e) /\ - ((EdgeSet.In b s /\ ~Interfere (fst_ext b) (snd_ext b) (merge e g p q)) \/ - EdgeSet.In b (VertexSet.fold - (fun y s' => EdgeSet.add (fst_ext e, y, Some N0) s') - (preference_adj (fst_ext e) (merge e g p q)) - EdgeSet.empty)))). - -Proof. -intros e g s p q; split; intros. -unfold AE_merge_up in H0. -set (f := (fun (y : VertexSet.elt) (s'' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s'')) in *. -set (diff1 := (VertexSet.diff (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g))) in *. -set (diff2 := (VertexSet.diff (preference_adj (fst_ext e) g) - (preference_adj (fst_ext e) (merge e g p q)))) in *. -cut (get_weight b = Some N0). intro Hw. -cut (~incident b (snd_ext e)). intro Hcut. -split. -assumption. -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH diff1). clear HH. intro HH. -induction (VertexSet.elements diff1). simpl in H0. -left. split. -apply (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0)). -intro H1. unfold Interfere in H1. -generalize (merge_4 e (fst_ext b, snd_ext b, None) _ p q H1). intro H2. -destruct H2. destruct H2. -unfold redirect in H3. -destruct (OTFacts.eq_dec (fst_ext x) (snd_ext e)). destruct H3. -assert (eq (fst_ext b, snd_ext b, None) (fst_ext e, snd_ext x, get_weight x)). -unfold weak_eq in H3. change_rewrite. destruct H3. -apply eq_ordered_eq; split; simpl; auto. -unfold same_type in H4. destruct H4. -destruct H4. destruct H5. simpl in H5. inversion H5. -destruct H4. rewrite <-H4. apply OptionN_as_OT.eq_refl. -destruct H3. -unfold same_type in H4. destruct H4. -destruct H4. destruct H6. simpl in H6. inversion H6. -destruct H4. rewrite <-H4. rewrite edge_comm. apply eq_ordered_eq. -split; auto. simpl. apply OptionN_as_OT.eq_refl. -generalize H5. clear H3 H4 H5. intro H3. -destruct (eq_charac _ _ H3); destruct H4; change_rewrite. -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, snd_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. destruct H7. -rewrite <-H5 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H4 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H5 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H4. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro H9. -rewrite <-H9. assumption. -rewrite edge_comm. rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, snd_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. -destruct H7. -rewrite <-H4 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H5 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H4 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H5. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro H9. -rewrite edge_comm. rewrite <-H9. assumption. -rewrite edge_comm. rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -rewrite edge_comm. apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -destruct (OTFacts.eq_dec (snd_ext x) (snd_ext e)). -assert (eq (fst_ext b, snd_ext b, None) (fst_ext x, fst_ext e, get_weight x)). -destruct H3. unfold same_type in H4. destruct H4; destruct H4. -destruct H5. simpl in H5. inversion H5. rewrite <-H4. -unfold weak_eq in H3. change_rewrite. -destruct H3; destruct H3. -apply eq_ordered_eq. split; auto. simpl. apply OptionN_as_OT.eq_refl. -rewrite edge_comm. apply eq_ordered_eq. split; auto. simpl. apply OptionN_as_OT.eq_refl. -generalize H4. clear H3 H4. intro H3. -destruct (eq_charac _ _ H3); change_rewrite. destruct H4. -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, fst_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. -destruct H7. -rewrite <-H4 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H5 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H4 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H5. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro H9. -rewrite <-H9. assumption. -rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -rewrite edge_comm. apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -destruct H4. -generalize (not_incident_merge_1 _ _ _ _ H0). intro H6. -cut (eq b (fst_ext e, fst_ext x, Some N0)). intro H7. -rewrite H7 in H6. clear H7. -generalize (not_incident_merge_2 _ _ _ _ H6). intro H7. -unfold diff2 in H7. -generalize (not_in_diff_equiv _ _ _ H7). clear H7. intro H7. -destruct H7. -rewrite <-H5 in H7. generalize (sym_pmap_merge_map e g q p _ _ H7). intro H8. -rewrite <-H4 in H8. generalize (sym_pmap_merge_map e g q p _ _ H8). intro H9. -elim (simple_graph (merge e g p q) (snd_ext b) (fst_ext b)). split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -destruct H1. -generalize (AE_weights _ _ H1). simpl. congruence. -assumption. -assumption. -rewrite <-H5 in H7. -elim H7. apply (sym_pmap g). -rewrite <-H4. -cut (In_graph_edge b g). -intro H8. -destruct H8. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro H9. -rewrite edge_comm. rewrite <-H9. assumption. -rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -generalize (IE_weights _ _ H8). congruence. -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -apply eq_ordered_eq. -constructor; simpl. split. auto. auto. -fold (get_weight b). rewrite Hw. apply OptionN_as_OT.eq_refl. - -assert (eq (fst_ext b, snd_ext b, None) x). -destruct H3. unfold same_type in H4. destruct H4; destruct H4. -destruct H5. simpl in H5. inversion H5. -rewrite <-H4. -unfold weak_eq in H3; change_rewrite. destruct H3; destruct H3. -apply eq_ordered_eq; split; auto. apply OptionN_as_OT.eq_refl. -rewrite edge_comm. apply eq_ordered_eq; split; auto. apply OptionN_as_OT.eq_refl. -generalize H4. clear H3 H4. intro H3. -elim (simple_graph g (fst_ext b) (snd_ext b)). -split. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap g))). -rewrite edge_comm. rewrite H3. -destruct H2. -rewrite <-H3 in H2. generalize (AE_weights _ _ H2). simpl. congruence. -assumption. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -assert (In_graph_edge b g). -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intuition. -destruct H4. -assert (eq b (fst_ext b, snd_ext b, Some N0)). -rewrite (edge_eq b). change_rewrite. -rewrite Hw. apply eq_refl. -rewrite edge_comm. rewrite <-H5. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite MEdgeFacts.fold_left_assoc in H0. -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -set (tmp := fold_left f' l set) in *. -unfold f' in H0. unfold f in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (fst_ext e, a, Some N0) b) in H1. -right. -rewrite <-H1. -rewrite VertexSet.fold_1. -fold f'. -set (s' := preference_adj (fst_ext e) (merge e g p q)) in *. -generalize VertexSet.elements_1. intro HHH. -generalize (HHH s' a). clear HHH. intro HHH. -induction (VertexSet.elements s'). simpl. -assert (InA Vertex.eq a nil). -apply HHH. -unfold s'. -assert (VertexSet.In a diff1). -apply HH. left. intuition. -apply (VertexSet.diff_1 H2). inversion H2. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left f' l0 EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec a a0). -apply EdgeSet.add_1. -apply eq_ordered_eq. constructor;simpl. -split; intuition. -apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl0. intros. -generalize (HHH H2). intro. -inversion H3; subst. -elim (n H5). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -apply IHl. -assumption. -intros. -apply HH. -right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_2. intro HH. -generalize (HH diff1). clear HH. intro HH. -induction (VertexSet.elements diff1). simpl in H0. -intro Hinc. destruct Hinc. -assert (eq b (snd_ext e, snd_ext b, Some N0)). -rewrite (edge_eq b). change_rewrite. rewrite Hw. -apply eq_ordered_eq. -constructor;simpl. split;intuition. -apply OptionN_as_OT.eq_refl. - -rewrite H2 in H0. -generalize (not_incident_merge_2 _ _ _ _ H0). intro H3. -elim H3. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). rewrite <-H2. -assert (In_graph_edge b g). -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intro. destruct H4. rewrite <-H2 in H5. assumption. -destruct H4. -assumption. -generalize (IE_weights _ _ H4). congruence. - -assert (eq b (fst_ext b, snd_ext e, Some N0)). -rewrite (edge_eq b). change_rewrite. rewrite Hw. -apply eq_ordered_eq. -constructor;simpl. split; intuition. -apply OptionN_as_OT.eq_refl. - -rewrite H2 in H0. rewrite edge_comm in H0. -generalize (not_incident_merge_2 _ _ _ _ H0). intro H3. -elim H3. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). rewrite edge_comm. rewrite <-H2. -assert (In_graph_edge b g). -generalize (proj1 (H _) (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0))). -intro. destruct H4. rewrite edge_comm in H2. rewrite <-H2 in H5. assumption. -destruct H4. -assumption. -generalize (IE_weights _ _ H4). congruence. - -rewrite MEdgeFacts.fold_left_assoc in H0. -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -set (tmp := fold_left f' l set) in *. -unfold f' in H0. unfold f in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (fst_ext e, a, Some N0) b) in H1. -intro. destruct H2. -destruct (eq_charac _ _ H1); change_rewrite. destruct H3. -elim (In_graph_edge_diff_ext _ _ p). -apply Vertex.eq_trans with (y := fst_ext b); auto. -destruct H3. -assert (VertexSet.In a diff1). apply HH. -left. intuition. -unfold diff1 in H5. generalize (VertexSet.diff_2 H5). clear H5. intro H5. -elim H5. -rewrite H4. rewrite <-H2. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -generalize p. intro Hin. -destruct Hin. -assert (eq (fst_ext e, snd_ext e, Some N0) e). -apply eq_ordered_eq. -constructor. simpl. split; apply Regs.eq_refl. -simpl. fold (get_weight e). rewrite (AE_weights _ _ H6). -apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -generalize (IE_weights _ _ H6). generalize q. intro Haff. destruct Haff. congruence. - -destruct (eq_charac _ _ H1); change_rewrite. destruct H3. -assert (VertexSet.In a diff1). apply HH. -left. intuition. -unfold diff1 in H5. generalize (VertexSet.diff_2 H5). clear H5. intro H5. -elim H5. -rewrite H4. rewrite <-H2. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). -generalize p. intro Hin. -destruct Hin. -assert (eq (fst_ext e, snd_ext e, Some N0) e). -apply eq_ordered_eq. -constructor. simpl. split; apply Regs.eq_refl. -simpl. fold (get_weight e). rewrite (AE_weights _ _ H6). -apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -generalize (IE_weights _ _ H6). generalize q. intro Haff. destruct Haff. congruence. -elim (In_graph_edge_diff_ext _ _ p). -apply Vertex.eq_trans with (y := snd_ext b); auto. -destruct H3. auto. -apply IHl. auto. -intros. apply HH. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -rewrite VertexSet.fold_1 in H0. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements diff1). simpl in H0. -assert (aff_edge b /\ In_graph_edge b g). -rewrite <-H. -apply (not_incident_merge_1 _ _ _ _ (not_incident_merge_1 _ _ _ _ H0)). -destruct H1. -destruct H2. -apply (AE_weights _ _ H2). -destruct H1. generalize (IE_weights _ _ H2). congruence. - -rewrite MEdgeFacts.fold_left_assoc in H0. -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -set (tmp := fold_left f' l set) in *. -unfold f' in H0. unfold f in H0. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H0). -fold (eq (fst_ext e, a, Some N0) b) in H1. -rewrite <-H1. auto. -apply IHl. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -destruct H0. -destruct H1. -destruct H1. -unfold AE_merge_up. -set (f := (fun (y : VertexSet.elt) (s'' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s'')) in *. -set (diff1 := (VertexSet.diff (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g))) in *. -set (diff2 := (VertexSet.diff (preference_adj (fst_ext e) g) - (preference_adj (fst_ext e) (merge e g p q)))) in *. -cut (EdgeSet.In b (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))). intro H3. -rewrite VertexSet.fold_1. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements diff1). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (set := (fold_left f' l - (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g)))) in *. -unfold f'. unfold f. -apply EdgeSet.add_2. assumption. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -cut (EdgeSet.In b (not_incident_merge (fst_ext e) s diff2)). intro. -set (set := (not_incident_merge (fst_ext e) s diff2)) in *. -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (snd_ext e, y, Some 0%N) s')) in *. -set (adjsnd := preference_adj (snd_ext e) g) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -induction (VertexSet.elements adjsnd). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left h' l set) in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. elim H0. -destruct (eq_charac _ _ H4); change_rewrite; destruct H5. -left. auto. -right. auto. assumption. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (fst_ext e, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -generalize VertexSet.elements_2. intro. -generalize (H3 diff2). clear H3. intro HH. -induction (VertexSet.elements diff2). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left h' l s) in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. -assert (VertexSet.In a diff2). -apply HH. left. intuition. -unfold diff2 in H4. -elim (VertexSet.diff_2 H4). clear H4. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -assert (eq (fst_ext e, a, Some N0) b). auto. generalize H4. clear H3 H4. intro H3. -rewrite H3. -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro. -rewrite H4. -cut (In_graph_edge (fst_ext b, snd_ext b, Some N0) (merge e g p q)). intro. -destruct H5. assumption. -generalize (IE_weights _ _ H5). simpl. congruence. -rewrite <-H4. -cut (eq b (redirect (snd_ext e) (fst_ext e) b)). intro. -rewrite H5. rewrite (edge_eq _). -cut (get_weight (redirect (snd_ext e) (fst_ext e) b) = Some N0). intro. -rewrite H6. -assert (exists w, In_graph_edge - (fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), Some w) (merge e g p q)). -apply merge_5. - -rewrite H in H1. intuition. -intro. -elim H0. -destruct (eq_charac _ _ H7);change_rewrite; destruct H8. -right. auto. -left. auto. -unfold aff_edge. exists N0. rewrite <-H3. auto. - -cut (eq (fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), - None) (fst_ext b, snd_ext b, None)). intro. -unfold Interfere. rewrite H7. assumption. - -unfold redirect. destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H0. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H0. right. auto. -apply eq_refl. - -destruct H7. -destruct H7. generalize (AE_weights _ _ H7). simpl. intro. -rewrite H8 in H7. left. auto. -generalize (IE_weights _ _ H7). simpl. congruence. - -rewrite <-H5. rewrite H4. auto. - -unfold redirect. destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H0. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H0. right. auto. -apply eq_refl. - -apply eq_ordered_eq. constructor; simpl; try split. -apply Regs.eq_refl. -apply Regs.eq_refl. -fold (get_weight b). rewrite <-H3. simpl. apply OptionN_as_OT.eq_refl. - -apply IHl. intros. apply HH. right. auto. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -(* last part !!!!!!! *) - -unfold AE_merge_up. -set (f := (fun (y : VertexSet.elt) (s'' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s'')) in *. -set (diff1 := (VertexSet.diff (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g))) in *. -set (diff2 := (VertexSet.diff (preference_adj (fst_ext e) g) - (preference_adj (fst_ext e) (merge e g p q)))) in *. - -set (pref := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1 in H1. -generalize VertexSet.elements_2. intro HH. -generalize (HH pref). clear HH. intro HH. -set (f' := fun a e => f e a) in *. induction (VertexSet.elements pref). -simpl in H1. elim (EdgeSet.empty_1 H1). - -set (set := (not_incident_merge (snd_ext e) - (not_incident_merge (fst_ext e) s diff2) - (preference_adj (snd_ext e) g))) in *. -rewrite MEdgeFacts.fold_left_assoc in H1. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f' in H1. unfold f in H1. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H1). -fold (eq (fst_ext e, a, Some N0) b) in H2. -rewrite <-H2. -destruct (Props.In_dec a (preference_adj (fst_ext e) g)). -cut (EdgeSet.In b set). intro Hcut. -rewrite VertexSet.fold_1. fold f'. -assert (~InA Vertex.eq a (VertexSet.elements diff1)). -intro. generalize (VertexSet.elements_2 H3). intro H4. -unfold diff1 in H4. elim (VertexSet.diff_2 H4 i). -induction (VertexSet.elements diff1). simpl. -rewrite H2. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left f' l0 set) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec a a0). -apply EdgeSet.add_1. apply eq_ordered_eq. constructor; simpl. split; intuition. -apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl0. intro. elim H3. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -unfold set. cut (EdgeSet.In b (not_incident_merge (fst_ext e) s diff2)). intro. -set (set' := (not_incident_merge (fst_ext e) s diff2)) in *. -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (snd_ext e, y, Some 0%N) s')) in *. -set (adjsnd := preference_adj (snd_ext e) g) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -induction (VertexSet.elements adjsnd). simpl. assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left h' l set') in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. elim H0. -destruct (eq_charac _ _ H4); change_rewrite; destruct H5. -left. auto. -right. auto. assumption. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H4)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H4). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H4)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -unfold not_incident_merge. -set (h := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.remove (fst_ext e, y, Some 0%N) s')) in *. -rewrite VertexSet.fold_1. -set (h' := fun a e => h e a) in *. -generalize VertexSet.elements_2. intro. -generalize (H3 diff2). clear H3. intro HHH. -induction (VertexSet.elements diff2). simpl. -rewrite H. split. -unfold aff_edge. exists N0. rewrite <-H2. auto. -left. rewrite <-H2. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap g))). assumption. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left h' l0 s) in *. -unfold h'. unfold h. -apply EdgeSet.remove_2. -intro. -assert (Vertex.eq a a0). -destruct (Vertex.eq_dec a a0). intuition. -assert (eq (fst_ext e, a0, Some N0) b) by auto. generalize H4. clear H3 H4. intro H3. -rewrite <-H3 in H2. destruct (eq_charac _ _ H2); change_rewrite; destruct H4. -auto. -apply Vertex.eq_trans with (y := fst_ext e); auto. -assert (~VertexSet.In a diff2). -intro. unfold diff2 in H5. elim (VertexSet.diff_2 H5). -apply HH. left. intuition. -elim H5. -rewrite H4. apply HHH. left. intuition. -apply IHl0. -intros. apply HHH. right. auto. - -unfold h'. unfold h. intros. -split; intros. -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 (EdgeSet.remove_3 H3)). -apply EdgeSet.remove_2. -intro H5. elim (EdgeSet.remove_1 H5 H3). -apply (EdgeSet.remove_3 (EdgeSet.remove_3 H3)). - -unfold h'. unfold h. intros. -apply RegRegProps.Equal_remove. auto. - -rewrite VertexSet.fold_1. -fold f'. -generalize VertexSet.elements_1. intro HHH. -generalize (HHH diff1 a). clear HHH. intro HHH. -assert (VertexSet.In a diff1). apply VertexSet.diff_3. apply HH. left. intuition. -assumption. -induction (VertexSet.elements diff1). simpl. -generalize (HHH H3). intro H4. inversion H4. -rewrite MEdgeFacts.fold_left_assoc. -set (tmp' := fold_left f' l0 set) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec a a0). -apply EdgeSet.add_1. -apply eq_ordered_eq. constructor; simpl; try split; intuition. -apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. apply IHl0. intro. -generalize (HHH H4). intro H5. inversion H5; subst. -elim (n0 H7). -assumption. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -apply IHl. -assumption. -intros. apply HH. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. -Qed. - -Lemma AE_aux_3 : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(forall b, EdgeSet.In b (AE (merge e g p q)) - <-> - (~incident b (snd_ext e) /\ - ((EdgeSet.In b s /\ ~Interfere (fst_ext b) (snd_ext b) (merge e g p q)) \/ - EdgeSet.In b (VertexSet.fold - (fun y s' => EdgeSet.add (fst_ext e, y, Some N0) s') - (preference_adj (fst_ext e) (merge e g p q)) - EdgeSet.empty)))). - -Proof. -intros e g s p q. generalize I. intro H. -split; intros. -generalize (proj1 (In_graph_aff_edge_in_AE _ _) H1). intro H3. -destruct H3 as [H3 H4]. -assert (~incident b (snd_ext e)) as Hinc. -intro. destruct H2; elim (merge_1 e g p q); auto. -unfold In_graph. rewrite H2. apply (proj1 (In_graph_edge_in_ext _ _ H4)). -unfold In_graph. rewrite H2. apply (proj2 (In_graph_edge_in_ext _ _ H4)). - -split. -assumption. - -assert (get_weight b = Some N0) as Hw. -apply AE_weights with (g := merge e g p q). assumption. - -clear H1. -generalize (merge_4 e _ g p q H4). intro H1. -destruct H1. destruct H1 as [H1 H2]. destruct H2 as [H2 HH2]. -assert (aff_edge x) as Haffb. -unfold same_type in HH2. destruct HH2; destruct H5;[auto|congruence]. - -unfold redirect in H2. -destruct (OTFacts.eq_dec (fst_ext x) (snd_ext e)). -right. -unfold weak_eq in H2. change_rewrite. destruct H2; destruct H2. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (snd_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (snd_ext x) nil). -apply HH. -unfold set. rewrite <-H5. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H2)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply eq_refl. -inversion H6. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (snd_ext x) a). -apply EdgeSet.add_1. -apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite <-e0. intuition. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; intuition. -rewrite H6. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H6). intro H7. -inversion H7; subst. -elim (n H9). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (snd_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (snd_ext x) nil). -apply HH. -unfold set. rewrite <-H2. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H5)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply edge_comm. -inversion H6. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (snd_ext x) a). -apply EdgeSet.add_1. -fold (eq (fst_ext e, a, Some N0) b). -rewrite edge_comm. apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite <-e0. intuition. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H6). intro H7. -inversion H7; subst. -elim (n H9). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -destruct (OTFacts.eq_dec (snd_ext x) (snd_ext e)). - -right. -unfold weak_eq in H2. change_rewrite. destruct H2; destruct H2. -assert (get_weight x = Some N0). -apply AE_weights with (g:=g). apply (proj2 (In_graph_aff_edge_in_AE _ _)). -split; auto. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (fst_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (fst_ext x) nil). -apply HH. -unfold set. rewrite <-H2. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H5)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (snd_ext b, fst_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H7. apply edge_comm. -inversion H7. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (fst_ext x) a). -apply EdgeSet.add_1. -fold (eq (fst_ext e, a, Some N0) b). -rewrite edge_comm. apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite <-e0. intuition. -rewrite Hw. auto. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H7). intro H8. -inversion H8; subst. -elim (n0 H10). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (set := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1. set (f' := fun a e => f e a) in *. -generalize VertexSet.elements_1. intro HH. -generalize (HH set (fst_ext x)). clear HH. intro HH. -induction (VertexSet.elements set). simpl. -assert (InA Vertex.eq (fst_ext x) nil). -apply HH. -unfold set. rewrite <-H5. rewrite (compat_preference_adj _ _ _ (Vertex.eq_sym H2)). -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -cut (eq b (fst_ext b, snd_ext b, Some N0)). intro Heq. -rewrite <-Heq. destruct H4. assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b); change_rewrite. -assert (get_weight b = Some N0). -apply AE_weights with (g:= merge e g p q). rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply eq_refl. -inversion H6. - -rewrite MEdgeFacts.fold_left_assoc. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f'. unfold f. -destruct (Vertex.eq_dec (fst_ext x) a). -apply EdgeSet.add_1. -apply eq_ordered_eq. -rewrite (edge_eq b). -constructor; simpl. split; intuition. -rewrite H5. intuition. -rewrite Hw. apply OptionN_as_OT.eq_refl. -apply EdgeSet.add_2. -apply IHl. intro. generalize (HH H6). intro H7. -inversion H7; subst. -elim (n0 H9). -auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -assert (eq b x) as Heq. -unfold weak_eq in H2. destruct H2; destruct H2. -apply eq_ordered_eq. constructor; simpl. -inversion H2; auto. -fold (get_weight b). fold (get_weight x). -rewrite Hw. -assert (get_weight x = Some N0). -apply AE_weights with (g:=g). -rewrite In_graph_aff_edge_in_AE. -split; auto. -rewrite H6. apply OptionN_as_OT.eq_refl. -rewrite (edge_eq b). rewrite (edge_eq x). -rewrite edge_comm. apply eq_ordered_eq. constructor; simpl. -inversion H2; simpl in *; intuition. -rewrite H6. rewrite H7. auto. -rewrite H6. rewrite H7. auto. -assert (get_weight x = Some N0). -apply AE_weights with (g:=g). -rewrite In_graph_aff_edge_in_AE. -split; intuition. -rewrite H6. rewrite Hw. apply OptionN_as_OT.eq_refl. -left. split. -rewrite Heq. -rewrite (H0 x). split. -exists N0. rewrite <-Heq. auto. auto. -intro. elim (simple_graph (merge e g p q) (fst_ext b) (snd_ext b)). -split. -unfold Interfere in H5. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ None (sym_imap_merge_map e g q p))). -rewrite edge_comm. -destruct H5. -generalize (AE_weights _ _ H5). simpl. congruence. -assumption. -apply (proj1 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -rewrite edge_comm. -cut (eq (fst_ext b, snd_ext b, Some N0) b). intro Heq2. -rewrite Heq2. -destruct H4. -assumption. -generalize (IE_weights _ _ H4). congruence. -rewrite (edge_eq b). change_rewrite. rewrite Hw. apply eq_refl. - -(* second part !!! *) - -destruct H1. -cut (get_weight b = Some N0). intro Hw. -assert (eq b (fst_ext b, snd_ext b, Some N0)). -rewrite (edge_eq b); change_rewrite. -apply eq_ordered_eq. constructor; simpl. -split; intuition. -rewrite Hw. apply OptionN_as_OT.eq_refl. - -destruct H2. -rewrite H3. -rewrite In_graph_aff_edge_in_AE. split. -exists N0; simpl; auto. -cut (eq b (redirect (snd_ext e) (fst_ext e) b)). intro Hcut. -rewrite <-H3. rewrite Hcut. rewrite (edge_eq _). -cut (get_weight (redirect (snd_ext e) (fst_ext e) b) = Some N0). intro Hw'. -cut (Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) b)) - (snd_ext (redirect (snd_ext e) (fst_ext e) b)) - (merge e g p q)). intro. -unfold Prefere in H4. destruct H4. -assert (get_weight ((fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), Some x)) = Some N0). -apply (AE_weights) with (g:=merge e g p q). -rewrite In_graph_aff_edge_in_AE. split. -exists x. auto. auto. -rewrite Hw'. rewrite <-H5. simpl. auto. -destruct H2. apply merge_5. -apply (proj1 (H0 _) H2). -intro. elim H1. -destruct (eq_charac _ _ H5). right; intuition. left; intuition. -exists N0; simpl; auto. -intro. unfold Interfere in H5. -cut (eq (fst_ext (redirect (snd_ext e) (fst_ext e) b), - snd_ext (redirect (snd_ext e) (fst_ext e) b), - None) - (fst_ext b, snd_ext b, None)). intro H6. -rewrite H6 in H5. -elim H4. unfold Interfere. assumption. -unfold redirect; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H1. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H1. right. auto. -apply eq_refl. -rewrite <-Hcut. assumption. -unfold redirect; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext b) (snd_ext e)). -elim H1. left. auto. -destruct (OTFacts.eq_dec (snd_ext b) (snd_ext e)). -elim H1. right. auto. -apply eq_refl. - -rewrite H3. -apply (proj2 (edgemap_to_edgeset_charac _ _ _ (Some N0) (sym_pmap_merge_map e g q p))). -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (s' := preference_adj (fst_ext e) (merge e g p q)) in *. -rewrite VertexSet.fold_1 in H2. -generalize VertexSet.elements_2. intro HH. -generalize (HH s'). clear HH. intro HH. -induction (VertexSet.elements s'). simpl in H2. -elim (EdgeSet.empty_1 H2). -set (f' := fun a e => f e a) in *. -rewrite MEdgeFacts.fold_left_assoc in H2. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f' in H2. unfold f in H2. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H2). -fold (eq (fst_ext e, a, Some N0) b) in H4. -rewrite (edge_eq b) in H4. -destruct (eq_charac _ _ H4); change_rewrite. -destruct H5. -assert (VertexSet.In (snd_ext b) s'). -apply HH. left. intuition. -unfold s' in H7. -unfold adj_set. rewrite (MapFacts.find_o _ (Vertex.eq_sym H5)). -assumption. -destruct H5. -assert (VertexSet.In (fst_ext b) s'). -apply HH. left. intuition. -unfold s' in H7. -unfold preference_adj in H7. unfold adj_set in H7. rewrite (MapFacts.find_o _ H5) in H7. -apply (sym_pmap_merge_map e g q p). assumption. -apply IHl. -assumption. -intros. apply HH. right. auto. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. - -destruct H2. -rewrite AE_weights with (g:=g). auto. -rewrite (In_graph_aff_edge_in_AE). rewrite <-H0. intuition. -set (f := (fun (y : VertexSet.elt) (s' : EdgeSet.t) => - EdgeSet.add (fst_ext e, y, Some 0%N) s')) in *. -set (s' := (preference_adj (fst_ext e) (merge e g p q))) in *. -rewrite VertexSet.fold_1 in H2. -set (f' := fun a e => f e a) in *. -induction (VertexSet.elements s'). simpl in H2. -elim (EdgeSet.empty_1 H2). -rewrite MEdgeFacts.fold_left_assoc in H2. -set (tmp := fold_left f' l EdgeSet.empty) in *. -unfold f' in H2. unfold f in H2. -destruct (proj1 (RegRegProps.Dec.F.add_iff _ _ _) H2). -fold (eq (fst_ext e, a, Some N0) b) in H3. rewrite <-H3. -auto. -apply IHl. assumption. - -unfold f'. unfold f. intros. -apply RegRegProps.add_add. - -unfold f'. unfold f. intros. -apply RegRegProps.Dec.F.add_m. apply eq_refl. auto. -Qed. - -Lemma AE_merge_wl_aux : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(EdgeSet.Equal (AE_merge_up (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g) - (preference_adj (snd_ext e) g) - e s) - (AE (merge e g p q))). - -Proof. -intros. -unfold EdgeSet.Equal. intro. rewrite (AE_aux_2 e g s p q H). - rewrite (AE_aux_3 e g s p q H). -reflexivity. -Qed. - -Lemma AE_merge_wl : forall e g s p q, -(forall a, EdgeSet.In a s <-> aff_edge a /\ In_graph_edge a g) -> -(forall b, EdgeSet.In b (AE_merge_up (preference_adj (fst_ext e) (merge e g p q)) - (preference_adj (fst_ext e) g) - (preference_adj (snd_ext e) g) - e s) - <-> - aff_edge b /\ In_graph_edge b (merge e g p q)). - -Proof. -intros. -rewrite <-(In_graph_aff_edge_in_AE). -apply AE_merge_wl_aux; assumption. -Qed. - -(* It implements the interface *) - -(* Specification of the interference neighborhood *) -Lemma in_interf : forall x y g, -VertexSet.In x (interference_adj y g) <-> In_graph_edge (x,y,None) g. - -Proof. -split; intros. -apply in_interf_interf. auto. -apply interf_in_interf. auto. -Qed. - -(* Specification of the preference neighborhood *) -Lemma in_pref : forall x y g, -VertexSet.In x (preference_adj y g) <-> exists w, In_graph_edge (x,y,Some w) g. - -Proof. -split; intros. -apply in_pref_pref. auto. -destruct H. apply pref_in_pref with (w:=x0). auto. -Qed.
\ No newline at end of file diff --git a/backend/InterfGraphProperties.v b/backend/InterfGraphProperties.v deleted file mode 100755 index 09e1b59..0000000 --- a/backend/InterfGraphProperties.v +++ /dev/null @@ -1,8 +0,0 @@ -Require Import InterfGraph. -Require Import Coloring. - -Lemma set_reg_reg_diff_ext : forall x f live live0, -SetRegReg.In x (interf_reg_reg (interf_graph f live live0)) -> fst x <> snd x. - -Proof. -Admitted.
\ No newline at end of file diff --git a/backend/InterfGraph_Construction.v b/backend/InterfGraph_Construction.v deleted file mode 100755 index 083a80b..0000000 --- a/backend/InterfGraph_Construction.v +++ /dev/null @@ -1,189 +0,0 @@ -Require Import Coqlib. -Require Import FSets. -Require Import FSetAVL. -Require Import Maps. -Require Import Ordered. -Require Import Registers. -Require Import Locations. -Require Import AST. -Require Import Op. -Require Import RTLtyping. -Require Import RTL. -Require Import Conventions. -Require Import InterfGraph. - -Definition add_interf_live - (filter: reg -> bool) (res: reg) (live: Regset.t) (g: graph): graph := - Regset.fold - (fun r g => if filter r then add_interf r res g else g) live g. - -Definition add_interf_op - (res: reg) (live: Regset.t) (g: graph): graph := - add_interf_live - (fun r => if Reg.eq r res then false else true) - res live g. - -Definition add_interf_move - (arg res: reg) (live: Regset.t) (g: graph): graph := - add_interf_live - (fun r => - if Reg.eq r res then false else - if Reg.eq r arg then false else true) - res live g. - -Definition add_interf_destroyed - (live: Regset.t) (destroyed: list mreg) (g: graph): graph := - List.fold_left - (fun g mr => Regset.fold (fun r g => add_interf_mreg r mr g) live g) - destroyed g. - -Definition add_interfs_indirect_call - (rfun: reg) (locs: list loc) (g: graph): graph := - List.fold_left - (fun g loc => - match loc with R mr => add_interf_mreg rfun mr g | _ => g end) - locs g. - -Definition add_interf_call - (ros: reg + ident) (locs: list loc) (g: graph): graph := - match ros with - | inl rfun => add_interfs_indirect_call rfun locs g - | inr idfun => g - end. - -Fixpoint add_prefs_call - (args: list reg) (locs: list loc) (g: graph) {struct args} : graph := - match args, locs with - | a1 :: al, l1 :: ll => - add_prefs_call al ll - (match l1 with R mr => add_pref_mreg a1 mr g | _ => g end) - | _, _ => g - end. - -Definition add_interf_entry - (params: list reg) (live: Regset.t) (g: graph): graph := - List.fold_left (fun g r => add_interf_op r live g) params g. - -Fixpoint add_interf_params - (params: list reg) (g: graph) {struct params}: graph := - match params with - | nil => g - | p1 :: pl => - add_interf_params pl - (List.fold_left - (fun g r => if Reg.eq r p1 then g else add_interf r p1 g) - pl g) - end. - -Definition add_edges_instr - (sig: signature) (i: instruction) (live: Regset.t) (g: graph) : graph := - match i with - | Iop op args res s => - if Regset.mem res live then - match is_move_operation op args with - | Some arg => - add_pref arg res (add_interf_move arg res live g) - | None => - add_interf_op res live g - end - else g - | Iload chunk addr args dst s => - if Regset.mem dst live - then add_interf_op dst live g - else g - | Icall sig ros args res s => - let largs := loc_arguments sig in - let lres := loc_result sig in - add_prefs_call args largs - (add_pref_mreg res lres - (add_interf_op res live - (add_interf_call ros largs - (add_interf_destroyed - (Regset.remove res live) destroyed_at_call_regs g)))) - | Itailcall sig ros args => - let largs := loc_arguments sig in - add_prefs_call args largs - (add_interf_call ros largs g) - | Ireturn (Some r) => - add_pref_mreg r (loc_result sig) g - | _ => g - end. - -Definition add_edges_instrs (f: function) (live: PMap.t Regset.t) : graph := - PTree.fold - (fun g pc i => add_edges_instr f.(fn_sig) i live!!pc g) - f.(fn_code) - empty_graph. - -Definition interf_graph (f: function) (live: PMap.t Regset.t) (live0: Regset.t) := - add_prefs_call f.(fn_params) (loc_parameters f.(fn_sig)) - (add_interf_params f.(fn_params) - (add_interf_entry f.(fn_params) live0 - (add_edges_instrs f live))). - -(** * Graph coloring *) - -(** The actual coloring of the graph is performed by a function written - directly in Caml, and not proved correct in any way. This function - takes as argument the [RTL] function, the interference graph for - this function, an assignment of types to [RTL] pseudo-registers, - and the set of all [RTL] pseudo-registers mentioned in the - interference graph. It returns the coloring as a function from - pseudo-registers to locations. *) -(* -Parameter graph_coloring: - function -> graph -> regenv -> Regset.t -> (reg -> loc). -*) -(** To ensure that the result of [graph_coloring] is a correct coloring, - we check a posteriori its result using the following Coq functions. - Let [coloring] be the function [reg -> loc] returned by [graph_coloring]. - The three properties checked are: -- [coloring r1 <> coloring r2] if there is a conflict edge between - [r1] and [r2] in the interference graph. -- [coloring r1 <> R m2] if there is a conflict edge between pseudo-register - [r1] and machine register [m2] in the interference graph. -- For all [r] mentioned in the interference graph, - the location [coloring r] is acceptable and has the same type as [r]. -*) - -Definition check_coloring_1 (g: graph) (coloring: reg -> loc) := - SetRegReg.for_all - (fun r1r2 => - if Loc.eq (coloring (fst r1r2)) (coloring (snd r1r2)) then false else true) - g.(interf_reg_reg). - -Definition check_coloring_2 (g: graph) (coloring: reg -> loc) := - SetRegMreg.for_all - (fun r1mr2 => - if Loc.eq (coloring (fst r1mr2)) (R (snd r1mr2)) then false else true) - g.(interf_reg_mreg). - -Definition same_typ (t1 t2: typ) := - match t1, t2 with - | Tint, Tint => true - | Tfloat, Tfloat => true - | _, _ => false - end. - -Definition loc_is_acceptable (l: loc) := - match l with - | R r => - if In_dec Loc.eq l temporaries then false else true - | S (Local ofs ty) => - if zlt ofs 0 then false else true - | _ => - false - end. - -Definition check_coloring_3 (rs: Regset.t) (env: regenv) (coloring: reg -> loc) := - Regset.for_all - (fun r => - let l := coloring r in - andb (loc_is_acceptable l) (same_typ (env r) (Loc.type l))) - rs. - -Definition check_coloring - (g: graph) (env: regenv) (rs: Regset.t) (coloring: reg -> loc) := - andb (check_coloring_1 g coloring) - (andb (check_coloring_2 g coloring) - (check_coloring_3 rs env coloring)). diff --git a/backend/Interference_adjacency.v b/backend/Interference_adjacency.v deleted file mode 100755 index 3d5d90d..0000000 --- a/backend/Interference_adjacency.v +++ /dev/null @@ -1,66 +0,0 @@ -Require Import InterfGraphMapImp. -Require Import Graph_Facts. -Require Import FSets. -Require Import SetsFacts. -Require Import Edges. - -Import Edge RegFacts Props RegRegProps. - -(* Some properties about the interference adjacency - and the same about preference adjacency *) - -(* x is not in its own interference neighborhood *) -Lemma not_in_interf_self : forall x g, -~VertexSet.In x (interference_adj x g). - -Proof. -intros x g. rewrite in_interf. intro H. -elim (In_graph_edge_diff_ext _ _ H). auto. -Qed. - -(* x is not in its own preference neighborhood *) -Lemma not_in_pref_self : forall x g, -~VertexSet.In x (preference_adj x g). - -Proof. -intros x g. rewrite in_pref. intro H. destruct H. -elim (In_graph_edge_diff_ext _ _ H). auto. -Qed. - -(* If x is an interference neighbor of y in g - then y is an interference neighbor of x in g *) -Lemma interf_adj_comm : forall x y g, -VertexSet.In x (interference_adj y g) -> VertexSet.In y (interference_adj x g). - -Proof. -intros x y g H. rewrite in_interf. rewrite edge_comm. rewrite <-in_interf. auto. -Qed. - -(* If x is a preference neighbor of y in g - then y is a preference neighbor of x in g *) -Lemma pref_adj_comm : forall x y g, -VertexSet.In x (preference_adj y g) -> VertexSet.In y (preference_adj x g). - -Proof. -intros x y g H. -rewrite in_pref in H. destruct H. rewrite edge_comm in H. -rewrite in_pref. exists x0. assumption. -Qed. - -(* If x is an interference neighbor of any vertex of g then x is in g *) -Lemma in_interf_in : forall x r g, -VertexSet.In x (interference_adj r g) -> In_graph x g. - -Proof. -intros x r g H. rewrite in_interf in H. -apply (proj1 (In_graph_edge_in_ext _ _ H)). -Qed. - -(* If x is a preferenec neighbor of any vertex then x is in g *) -Lemma in_pref_in : forall x r g, -VertexSet.In x (preference_adj r g) -> In_graph x g. - -Proof. -intros x r g H. rewrite in_pref in H. destruct H. -apply (proj1 (In_graph_edge_in_ext _ _ H)). -Qed. diff --git a/backend/Merge_Adjacency.v b/backend/Merge_Adjacency.v deleted file mode 100755 index d6d879a..0000000 --- a/backend/Merge_Adjacency.v +++ /dev/null @@ -1,347 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Remove_Vertex_Degree. -Require Import Edges. -Require Import MyRegisters. -Require Import Interference_adjacency. -Require Import Graph_Facts. - -Module Register := Regs. - -Import Edge RegFacts Props. - -(* The following lemmas define the interference adjacency - of the first endpoint of e after the merge of e *) - -Lemma merge_interf_adj_fst_1 : forall e g H0 Haff, -VertexSet.Subset (interference_adj (fst_ext e) (merge e g H0 Haff)) - (VertexSet.union (interference_adj (fst_ext e) g) - (interference_adj (snd_ext e) g)). - -Proof. -unfold VertexSet.Subset. intros e g H Haff a H1. -rewrite in_interf in H1. generalize (In_merge_edge_inv _ _ _ H Haff H1); intro. -destruct H0 as [x H0]. destruct H0 as [H3 H4]. -destruct x as [ex wx]. destruct ex as [x1 x2]. -assert (interf_edge (x1,x2,wx)) as Hinterf. -destruct H4. unfold same_type in H2. destruct H2; destruct H2. -unfold aff_edge in H4. destruct H4. simpl in H4. congruence. -assumption. -destruct H4 as [H4 HH4]. -assert (eq (a, fst_ext e, None) (redirect (snd_ext e) (fst_ext e) (x1,x2,wx))). -apply weak_eq_interf_eq. assumption. unfold interf_edge. auto. -unfold interf_edge. rewrite redirect_weight_eq. auto. clear H4. -generalize (redirect_charac (x1,x2,wx) (snd_ext e) (fst_ext e)); intros. -destruct H2. destruct H2. destruct H4. rewrite <-H2 in H0. -apply VertexSet.union_2. rewrite in_interf. rewrite H0. assumption. -destruct H2. destruct H2. rewrite <-H2 in H0. change_rewrite. -apply VertexSet.union_3. rewrite in_interf. -destruct (eq_charac _ _ H0); destruct H5; change_rewrite. -assert (eq (a, snd_ext e, None) (x1, x2, wx)). -Eq_comm_eq. apply (Register.eq_trans _ _ _ H5 H6). -generalize (get_weight_m _ _ H0). simpl. intro. rewrite H7. apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -assert (eq (a,snd_ext e, None) (x1,x2,wx)). -Eq_comm_eq. -generalize (get_weight_m _ _ H0). simpl. intro. rewrite H7. apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -destruct H2; change_rewrite. rewrite <-H2 in H0. -destruct (eq_charac _ _ H0); destruct H5; change_rewrite. -apply VertexSet.union_3. rewrite in_interf. -assert (eq (a, snd_ext e, None) (x1,x2,wx)). -Eq_eq. -generalize (get_weight_m _ _ H0). simpl. intro. rewrite H7. apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -apply VertexSet.union_3. rewrite in_interf. -assert (eq (a, snd_ext e, None) (x1,x2,wx)). -Eq_eq. -apply (Register.eq_trans _ _ _ H5 H6). -generalize (get_weight_m _ _ H0). simpl. intro. rewrite H7. apply OptionN_as_OT.eq_refl. -rewrite H7. assumption. -Qed. - -Lemma merge_interf_adj_fst_2 : forall e g H0 Haff, -VertexSet.Subset (interference_adj (fst_ext e) g) - (interference_adj (fst_ext e) (merge e g H0 Haff)). - -Proof. -unfold VertexSet.Subset. intros. -rewrite in_interf in H. rewrite in_interf. -destruct (redirect_charac (a, fst_ext e, None) (snd_ext e) (fst_ext e)); intros. -change_rewrite. destruct H1. destruct H2. -rewrite H1. apply In_merge_interf_edge. assumption. -unfold interf_edge. auto. -change_rewrite. destruct H1. destruct H1. -elim (interf_pref_conflict (snd_ext e) (fst_ext e) g). -split. unfold Prefere. destruct Haff. exists x. -rewrite edge_comm. rewrite <-H3. rewrite (edge_eq e) in H0. assumption. -unfold Interfere. -assert (eq (snd_ext e, fst_ext e, None) (a, fst_ext e, None)) by Eq_eq. -rewrite H3. assumption. -destruct H1. rewrite H1. apply In_merge_interf_edge. assumption. -unfold interf_edge. auto. -Qed. - -Lemma merge_interf_adj_fst_3 : forall e g H0 Haff, -VertexSet.Subset (interference_adj (snd_ext e) g) - (interference_adj (fst_ext e) (merge e g H0 Haff)). - -Proof. -unfold VertexSet.Subset. intros. -rewrite in_interf in H. rewrite in_interf. -destruct (redirect_charac (a, snd_ext e, None) (snd_ext e) (fst_ext e)); intros. -change_rewrite. destruct H1. destruct H2. elim H3. auto. -change_rewrite. destruct H1. destruct H1. -elim (In_graph_edge_diff_ext _ _ H). change_rewrite. auto. -destruct H1. rewrite H1. apply In_merge_interf_edge. assumption. -unfold interf_edge. auto. -Qed. - -Lemma merge_interf_adj_fst : forall e g H0 Haff, -VertexSet.Equal (interference_adj (fst_ext e) (merge e g H0 Haff)) - (VertexSet.union (interference_adj (fst_ext e) g) - (interference_adj (snd_ext e) g)). - -Proof. -intros. split; intros. -apply (merge_interf_adj_fst_1 _ _ H0 Haff _ H). -destruct (VertexSet.union_1 H). - apply (merge_interf_adj_fst_2 _ _ H0 Haff _ H1). - apply (merge_interf_adj_fst_3 _ _ H0 Haff _ H1). -Qed. - -(* The following lemmas define the equality of the interference adjacency - of any vertex which is not an endpoint of e after the merge of e *) - -Lemma merge_interf_adj_1 : forall x e g Hin Haff, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -VertexSet.Subset (VertexSet.remove (snd_ext e) (interference_adj x g)) - (interference_adj x (merge e g Hin Haff)). - -Proof. -unfold VertexSet.Subset. intros. -rewrite in_interf. rewrite Dec.F.remove_iff in H1. destruct H1. -destruct (redirect_charac (a,x,None) (snd_ext e) (fst_ext e)); change_rewrite. -destruct H3. destruct H4. rewrite H3. -apply In_merge_interf_edge. rewrite <-in_interf. assumption. -unfold interf_edge. auto. -destruct H3. destruct H3. elim H2. auto. -destruct H3. elim H0. auto. -Qed. - -Lemma merge_interf_adj_2 : forall x e g Hin Haff, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -VertexSet.Subset (interference_adj x (merge e g Hin Haff)) - (VertexSet.add (fst_ext e) (interference_adj x g)). - -Proof. -unfold VertexSet.Subset. intros. -destruct (Register.eq_dec a (fst_ext e)); [apply VertexSet.add_1|apply VertexSet.add_2]; intuition. -rewrite in_interf in H1. generalize (In_merge_edge_inv _ _ _ Hin Haff H1). intro. -destruct H2. destruct H2. destruct H3. -assert (eq (a,x,None) (redirect (snd_ext e) (fst_ext e) x0)). -apply weak_eq_interf_eq. assumption. unfold interf_edge. auto. -unfold interf_edge. rewrite redirect_weight_eq. -unfold same_type in H4. destruct H4; destruct H4. -unfold aff_edge in H5. destruct H5. simpl in H5. congruence. -auto. intuition. -rewrite in_interf. rewrite H5. -destruct (redirect_charac x0 (snd_ext e) (fst_ext e)). -destruct H6. destruct H7. rewrite <-H6. assumption. -destruct H6. destruct H6. rewrite <-H6 in H5. -destruct (eq_charac _ _ H5); destruct H8; change_rewrite. -elim n. auto. elim H. auto. -destruct H6. rewrite <-H6 in H5. -destruct (eq_charac _ _ H5); destruct H8; change_rewrite. -elim H. auto. elim n. auto. -Qed. - -Lemma merge_interf_adj_not_in : forall x e g Hin Haff, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -~VertexSet.In x (interference_adj (snd_ext e) g) -> -VertexSet.Equal (interference_adj x (merge e g Hin Haff)) - (interference_adj x g). - -Proof. -intros. split; intros. -generalize (merge_interf_adj_2 x e g Hin Haff H H0 a H2). intro. -rewrite Dec.F.add_iff in H3. destruct H3. -rewrite <-H3 in H2. generalize (interf_adj_comm _ _ _ H2). intro. -rewrite merge_interf_adj_fst in H4. destruct (VertexSet.union_1 H4). -rewrite <-H3. apply interf_adj_comm. assumption. -elim H1. auto. -assumption. -apply merge_interf_adj_1; auto. -apply VertexSet.remove_2; auto. -intro. elim H1. apply interf_adj_comm. rewrite H3. assumption. -Qed. - -Lemma merge_interf_adj_in_snd : forall x e g Hin Haff, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -VertexSet.In x (interference_adj (snd_ext e) g) -> -VertexSet.Equal (interference_adj x (merge e g Hin Haff)) - (VertexSet.add (fst_ext e) (VertexSet.remove (snd_ext e) - (interference_adj x g))). - -Proof. -intros. split; intros. -generalize (merge_interf_adj_2 _ _ _ Hin Haff H H0 _ H2). intro. -rewrite Dec.F.add_iff in H3. destruct H3. -apply VertexSet.add_1. assumption. -apply VertexSet.add_2. apply VertexSet.remove_2. -intro. rewrite <-H4 in H2. rewrite in_interf in H2. -generalize (proj1 (In_graph_edge_in_ext _ _ H2)). change_rewrite. intro. -rewrite In_merge_vertex in H5. destruct H5. elim H6. auto. assumption. - -rewrite Dec.F.add_iff in H2. destruct H2. -rewrite <-H2. apply interf_adj_comm. rewrite merge_interf_adj_fst. -apply VertexSet.union_3. auto. -apply merge_interf_adj_1; auto. -Qed. - -Lemma merge_interf_adj_in_both : forall x e g Hin Haff, -VertexSet.In x (interference_adj (snd_ext e) g) -> -VertexSet.In x (interference_adj (fst_ext e) g) -> -VertexSet.Equal (interference_adj x (merge e g Hin Haff)) - (VertexSet.remove (snd_ext e) (interference_adj x g)). - -Proof. -intros. -assert (~Register.eq x (fst_ext e)). -intro. rewrite H1 in H0. elim (not_in_interf_self _ _ H0). -assert (~Register.eq x (snd_ext e)). -intro. rewrite H2 in H. elim (not_in_interf_self _ _ H). -rewrite merge_interf_adj_in_snd; auto. -split; intros. -rewrite Dec.F.add_iff in H3. destruct H3. -apply VertexSet.remove_2. intro. -elim (In_graph_edge_diff_ext _ _ Hin). rewrite H3. auto. -rewrite <-H3. apply interf_adj_comm. assumption. -assumption. -apply VertexSet.add_2. auto. -Qed. - -Lemma preference_adj_merge : forall x e g p q, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -VertexSet.Subset (preference_adj x (merge e g p q)) - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (preference_adj x g))). - -Proof. -intros x e g p q H0 H1. generalize I. intro H. -unfold VertexSet.Subset; intros. -rewrite in_pref in H2. destruct H2. -generalize (In_merge_edge_inv e _ g p q H2). intro. -destruct H3. destruct H3. destruct H4 as [H4 HH4]. -unfold weak_eq in H4. change_rewrite. destruct H4. -unfold redirect in H4; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)); change_rewrite. -destruct H4. apply VertexSet.add_1. intuition. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)); change_rewrite. -destruct H4. elim H0. intuition. -destruct H4. apply VertexSet.add_2. apply VertexSet.remove_2. -intro. elim n. rewrite H6. intuition. -rewrite in_pref. unfold same_type in HH4. -destruct HH4. destruct H6. -destruct H6. exists x2. -assert (eq (a,x,Some x2) x1). -apply eq_ordered_eq. split; try split; simpl; auto. -unfold get_weight in H6. rewrite H6. apply OptionN_as_OT.eq_refl. -rewrite H8. assumption. -destruct H6. unfold interf_edge in H7. simpl in H7. congruence. -unfold redirect in H4; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)); change_rewrite. -destruct H4. elim H0. auto. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)); change_rewrite. -destruct H4. apply VertexSet.add_1. intuition. -destruct H4. apply VertexSet.add_2. apply VertexSet.remove_2. -intro. elim n0. rewrite H6. intuition. -rewrite in_pref. unfold same_type in HH4. -destruct HH4. destruct H6. -destruct H6. exists x2. -assert (eq (a,x,Some x2) x1). -rewrite edge_comm. apply eq_ordered_eq. split; try split; simpl; auto. -unfold get_weight in H6. rewrite H6. apply OptionN_as_OT.eq_refl. -rewrite H8. assumption. -destruct H6. unfold interf_edge in H7. simpl in H7. congruence. -Qed. - -Lemma preference_adj_merge_2 : forall x e g p q, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -VertexSet.Subset (VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) (preference_adj x g))) - (preference_adj x (merge e g p q)). - -Proof. -unfold VertexSet.Subset; intros x e g p q H0 H1 a H2. generalize I. intro H. -assert (~Register.eq a (fst_ext e)). intro. -rewrite H3 in H2. elim (VertexSet.remove_1 (Register.eq_refl _) H2). -generalize (VertexSet.remove_3 H2). clear H2. intro H2. -assert (~Register.eq a (snd_ext e)). intro. -rewrite H4 in H2. elim (VertexSet.remove_1 (Register.eq_refl _) H2). -generalize (VertexSet.remove_3 H2). clear H2. intro. -rewrite in_pref in H2. destruct H2. -assert (exists w, In_graph_edge (a,x,Some w) (merge e g p q)). -cut (Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) (a,x,Some x0))) - (snd_ext (redirect (snd_ext e) (fst_ext e) (a,x,Some x0))) - (merge e g p q)). intro. -unfold redirect in H5; change_rewrite. -destruct (OTFacts.eq_dec a (snd_ext e)); change_rewrite. -elim (H4 r). -destruct (OTFacts.eq_dec x (snd_ext e)); change_rewrite. -elim (H1 r). -unfold Prefere in H5. assumption. -apply In_merge_pref_edge. assumption. -intro. destruct (eq_charac _ _ H5); change_rewrite; destruct H6. -elim H1. auto. -elim H0. auto. -unfold aff_edge. exists x0. auto. -intro. unfold redirect in H5;change_rewrite. -destruct (OTFacts.eq_dec a (snd_ext e)); change_rewrite. -elim (H4 r). -destruct (OTFacts.eq_dec x (snd_ext e)); change_rewrite. -elim (H1 r). -unfold Interfere in H5. generalize (In_merge_edge_inv e _ g p q H5). intro H8. -generalize I. intro H7. -destruct H8. destruct H6. destruct H8. -unfold weak_eq in H8. change_rewrite. destruct H8. -unfold redirect in H8; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)); change_rewrite; destruct H8. -elim (H3 H8). -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)); change_rewrite. -elim (H0 H10). -elim (interf_pref_conflict a x g). -split. -unfold Prefere. exists x0. assumption. -unfold Interfere. -assert (eq (a,x,None) x1). -apply weak_eq_interf_eq. unfold weak_eq. change_rewrite. -left. split; auto. unfold interf_edge. auto. -unfold same_type in H9. destruct H9. destruct H9. -unfold aff_edge in H11. destruct H11. simpl in H11. congruence. -destruct H9. assumption. -rewrite H11. assumption. - -unfold redirect in H8; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)); change_rewrite. -destruct H8. elim (H0 H10). -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)); change_rewrite. -destruct H8. elim (H3 H8). -elim (interf_pref_conflict a x g). -split. -unfold Prefere. exists x0. assumption. -unfold Interfere. destruct H8. -assert (eq (a,x,None) x1). -apply weak_eq_interf_eq. unfold weak_eq. change_rewrite. -right. split; auto. unfold interf_edge. auto. -unfold same_type in H9. destruct H9. destruct H9. -unfold aff_edge in H11. destruct H11. simpl in H11. congruence. -destruct H9. assumption. -rewrite H11. assumption. -rewrite in_pref. assumption. -Qed. diff --git a/backend/Merge_Degree.v b/backend/Merge_Degree.v deleted file mode 100755 index 1584979..0000000 --- a/backend/Merge_Degree.v +++ /dev/null @@ -1,205 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Merge_Adjacency. -Require Import ZArith. -Require Import Edges. -Require Import MyRegisters. -Require Import Merge_Adjacency. -Require Import Interference_adjacency. -Require Import Remove_Vertex_Degree. - -Module Register := Regs. - -Import Edge Props RegFacts. - -(* If x interferes with both (fst_ext e) and (snd_ext e), then - its degree decreases of one when e is coalesced *) -Lemma merge_degree_dec_inter : forall x e g Hin Haff, -VertexSet.In x (interference_adj (fst_ext e) g) -> -VertexSet.In x (interference_adj (snd_ext e) g) -> -interf_degree g x = S (interf_degree (merge e g Hin Haff) x). - -Proof. -intros. unfold interf_degree. -rewrite (cardinal_m (merge_interf_adj_in_both _ _ _ Hin Haff H0 H)). -rewrite remove_cardinal_1. reflexivity. apply interf_adj_comm. auto. -Qed. - -(* If x does not interfere with both (fst_ext e) and (snd_ext e) - then its degree is unchanged when e is coalesced *) -Lemma merge_dec_eq : forall x e g Hin Haff, -~VertexSet.In x (interference_adj (fst_ext e) g) \/ -~VertexSet.In x (interference_adj (snd_ext e) g) -> -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -interf_degree g x = interf_degree (merge e g Hin Haff) x. - -Proof. -intros. unfold interf_degree. destruct H. -destruct (In_dec x (interference_adj (snd_ext e) g)). -rewrite (cardinal_m (merge_interf_adj_in_snd _ _ _ Hin Haff H0 H1 i)). -rewrite add_cardinal_2. rewrite remove_cardinal_1. reflexivity. -apply interf_adj_comm. auto. -intro. elim H. apply interf_adj_comm. apply (VertexSet.remove_3 H2). -rewrite (cardinal_m (merge_interf_adj_not_in _ _ _ Hin Haff H0 H1 n)). reflexivity. -rewrite (cardinal_m (merge_interf_adj_not_in _ _ _ Hin Haff H0 H1 H)). reflexivity. -Qed. - -(* The interference degree of any vertex which is not an endpoint - of e decreases when e is coalesced *) -Lemma merge_degree_dec : forall x e g Hin Haff, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -interf_degree (merge e g Hin Haff) x <= interf_degree g x. - -Proof. -intros. destruct (In_dec x (interference_adj (fst_ext e) g)). -destruct (In_dec x (interference_adj (snd_ext e) g)). -rewrite (merge_degree_dec_inter x e g Hin Haff i i0). auto. -rewrite (merge_dec_eq x e g Hin Haff (or_intror _ n)); auto. -rewrite (merge_dec_eq x e g Hin Haff (or_introl _ n)); auto. -Qed. - -(* If x does not interfere with the first endpoint of e then - x is of low-degree in (merge e g Hin Haff) iff it is in g *) -Lemma low_merge_low_fst : forall x e g K Hin Haff, -~VertexSet.In x (interference_adj (fst_ext e) g) -> -~Register.eq x (snd_ext e) -> -~Register.eq x (fst_ext e) -> -has_low_degree (merge e g Hin Haff) K x = has_low_degree g K x. - -Proof. -intros x e g palette Hin Haff H H0 H1. unfold has_low_degree. -rewrite (merge_dec_eq x e g Hin Haff); auto. -Qed. - -(* If x does not interfere with the second endpoint of e then - x is of low-degree in (merge e g Hin Haff) iff it is in g *) -Lemma low_merge_low_snd : forall x e g K Hin Haff, -~VertexSet.In x (interference_adj (snd_ext e) g) -> -~Register.eq x (snd_ext e) -> -~Register.eq x (fst_ext e) -> -has_low_degree (merge e g Hin Haff) K x = has_low_degree g K x. - -Proof. -intros x e g palette Hin Haff H H0 H1. unfold has_low_degree. -rewrite (merge_dec_eq x e g Hin Haff); auto. -Qed. - -(* A high-degree vertex of (merge e g Hin Haff) is of high-degree in g *) -Lemma merge_low_1 : forall g e x K Haff Hin, -has_low_degree (merge e g Hin Haff) K x = false -> -~Register.eq x (snd_ext e) -> -~Register.eq x (fst_ext e) -> -has_low_degree g K x = false. - -Proof. -intros g e x K Haff Hin Hpre H0 H1. unfold has_low_degree in *. -destruct (le_lt_dec K (interf_degree (merge e g Hin Haff) x)); -destruct (le_lt_dec K (interf_degree g x )); inversion Hpre. -reflexivity. -generalize (merge_degree_dec x e g Hin Haff H1 H0). intro. -apply False_ind. intuition. -Qed. - -(* A low-degree vertex of g is of low-degree in (merge e g Haff Hin) *) -Lemma low_dec : forall x e g Hin Haff K, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -has_low_degree g K x = true -> -has_low_degree (merge e g Hin Haff) K x = true. - -Proof. -intros. -case_eq (has_low_degree (merge e g Hin Haff) K x);[auto|intros]. -rewrite (merge_low_1 g e x K Haff Hin H2 H0 H) in H1. inversion H1. -Qed. - -(* A vertex high-degree vertex of g (which is not an endpoint of e) - which is of low-degree in (merge e g p q) belongs to the interference - neighborhood of the two endpoints of e in g *) -Lemma merge_dec_interf : forall x e k g p q, -has_low_degree g k x = false -> -has_low_degree (merge e g p q) k x = true -> -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -VertexSet.In x (interference_adj (fst_ext e) g) /\ -VertexSet.In x (interference_adj (snd_ext e) g). - -Proof. -intros. -destruct (In_dec x (interference_adj (fst_ext e) g)). -split. assumption. -destruct (In_dec x (interference_adj (snd_ext e) g)). -assumption. -rewrite (low_merge_low_snd _ _ _ _ p q n H2 H1) in H0. rewrite H0 in H. inversion H. -rewrite (low_merge_low_fst _ _ _ _ p q n H2 H1) in H0. rewrite H0 in H. inversion H. -Qed. - -(* A vertex high-degree vertex of g (which is not an endpoint of e) - which is of low-degree in (merge e g p q) is of degree k in g *) -Lemma merge_dec_K : forall x e k g p q, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -has_low_degree g k x = false -> -has_low_degree (merge e g p q) k x = true -> -interf_degree g x = k. - -Proof. -intros x e k g p q H0 H1 H2 H3. generalize I. intro H. unfold interf_degree. -assert (VertexSet.In x (interference_adj (fst_ext e) g) /\ - VertexSet.In x (interference_adj (snd_ext e) g)). -apply (merge_dec_interf x e k g p q); auto. -destruct H4. -generalize (merge_degree_dec_inter x e g p q H4 H5). intro. -unfold has_low_degree, interf_degree in *. -destruct (le_lt_dec k (VertexSet.cardinal (interference_adj x (merge e g p q)))). -inversion H3. -destruct (le_lt_dec k (VertexSet.cardinal (interference_adj x g))). -rewrite H6 in l0. rewrite (lt_le_S_eq _ _ l l0). assumption. -inversion H2. -Qed. - -(* Reciprocally, a vertex of degree k interfering with - the two endpoints of g is of low-degree in (merge e g p q) *) -Lemma merge_dec_low : forall x e k g p q, -interf_degree g x = k -> -VertexSet.In x (interference_adj (fst_ext e) g) -> -VertexSet.In x (interference_adj (snd_ext e) g) -> -has_low_degree (merge e g p q) k x = true. - -Proof. -unfold interf_degree. intros x e k g p q H0 H1 H2. generalize I. intro H. -assert (~Register.eq x (fst_ext e)). -intro. elim (not_in_interf_self (fst_ext e) g). rewrite H3 in H1. assumption. -assert (~Register.eq x (snd_ext e)). -intro. elim (not_in_interf_self (snd_ext e) g). rewrite H4 in H2. assumption. -generalize (merge_degree_dec_inter x e g p q H1 H2). intro. -unfold has_low_degree, interf_degree in *. -destruct (le_lt_dec k (VertexSet.cardinal (interference_adj x (merge e g p q)))). -rewrite H5 in H0. rewrite <-H0 in l. elim (le_S_irrefl _ l). -reflexivity. -Qed. - -(* Again, unused but meaningful theorem, summarizing evolution of degree - when an edge e is coalesced *) - -Theorem merge_degree_evolution : forall x e k g p q, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -((has_low_degree g k x = false /\ has_low_degree (merge e g p q) k x = true) - <-> -(interf_degree g x = k /\ - VertexSet.In x (interference_adj (fst_ext e) g) /\ - VertexSet.In x (interference_adj (snd_ext e) g))). - -Proof. -split; intros. -destruct H1. -split. apply (merge_dec_K x e k g p q); auto. -apply (merge_dec_interf x e k g p q); auto. -destruct H1. destruct H2. -split. unfold has_low_degree. rewrite H1. -destruct (le_lt_dec k k). reflexivity. elim (lt_irrefl _ l). -apply merge_dec_low; auto. -Qed. diff --git a/backend/Merge_Move.v b/backend/Merge_Move.v deleted file mode 100755 index 1e20d00..0000000 --- a/backend/Merge_Move.v +++ /dev/null @@ -1,730 +0,0 @@ -Require Import FSetInterface. -Require Import InterfGraphMapImp. -Require Import Remove_Vertex_Move. -Require Import ZArith. -Require Import Edges. -Require Import MyRegisters. -Require Import Affinity_relation. -Require Import Interference_adjacency. -Require Import Graph_Facts. -Require Import Merge_Adjacency. - -Import Edge Props RegFacts. - -(* A nonmove-related vertex of g different from the first endpoint of e - is nonmove-related in (merge e g p q) *) -Lemma move_merge_false : forall x e g p q, -~Register.eq x (fst_ext e) -> -move_related g x = false -> -move_related (merge e g p q) x = false. - -Proof. -intros x e g p q Hfst H0. generalize I. intro H. -case_eq (move_related (merge e g p q) x); intros. -generalize (move_related_charac _ _ H1). intro. -destruct H2. destruct H2. destruct H3. -generalize (In_merge_edge_inv _ _ _ p q H3). intro. -destruct H5. destruct H5. -assert (move_related g x = true). -apply move_related_charac2 with (e := x1). -unfold weak_eq in H6. destruct H6. -unfold same_type in H7. destruct H7. destruct H7. assumption. -destruct H7. destruct H2. unfold interf_edge in H2. congruence. assumption. -destruct H6. -unfold redirect in H6. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)); destruct H6; change_rewrite; destruct H6. -destruct H4. -elim Hfst. rewrite H4. auto. -right. rewrite H4. auto. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)); change_rewrite. -destruct H4. -right. rewrite H4. auto. -elim Hfst. rewrite H4. auto. -destruct H4. -right. rewrite H4. auto. -elim Hfst. rewrite H4. auto. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)); change_rewrite. -destruct H4. -left. rewrite H4. auto. -elim Hfst. rewrite H4. auto. -destruct H4. -left. rewrite H4. auto. -right. rewrite H4. auto. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)); change_rewrite. -destruct H4. -elim Hfst. rewrite H4. auto. -left. rewrite H4. auto. -destruct H4. -right. rewrite H4. auto. -left. rewrite H4. auto. -rewrite H0 in H7. inversion H7. -auto. -Qed. - -(* Equivalently, a move-related vertex of (merge e g p q) is - move-related in g *) -Lemma move_related_merge_move_related : forall g e x H H0, -move_related (merge e g H H0) x = true -> -move_related g x = true. - -Proof. -intros. case_eq (move_related g x); auto. -intro. assert (~Register.eq x (fst_ext e)). -intro. rewrite (compat_bool_move _ _ _ H3) in H2. -rewrite (proj1 (Aff_edge_aff _ _ H H0)) in H2. inversion H2. -rewrite (move_merge_false _ _ _ H H0 H3 H2) in H1. inversion H1. -Qed. - -(* If x is neither a neighbor of (fst_ext e) nor (snd_ext e) and - if x is move-related in g, then x is move-related in (merge e g H Haff) *) -Lemma move_related_move_related_merge : forall g e x Haff H, -~VertexSet.In x (preference_adj (fst_ext e) g) -> -~VertexSet.In x (preference_adj (snd_ext e) g) -> -move_related g x = true -> -move_related (merge e g H Haff) x = true. - -Proof. -intros. -generalize (move_related_charac _ _ H2). intro. -destruct H3. destruct H3. destruct H4. -assert (~Register.eq x (fst_ext e)). -intro. elim H1. rewrite H6. -rewrite in_pref. destruct Haff. exists x1. rewrite <-H7. -rewrite (edge_eq e) in H. assumption. -assert (~Register.eq x (snd_ext e)). -intro. elim H0. rewrite H7. -rewrite in_pref. destruct Haff. exists x1. rewrite <-H8. -rewrite (edge_eq e) in H. rewrite edge_comm. assumption. - -assert (Prefere (fst_ext (redirect (snd_ext e) (fst_ext e) x0)) - (snd_ext (redirect (snd_ext e) (fst_ext e) x0)) - (merge e g H Haff)). -apply In_merge_pref_edge. assumption. -intro. destruct (eq_charac _ _ H8); destruct H9. -destruct H5. elim H6. rewrite H5. auto. -elim H7. rewrite H5. auto. -destruct H5. elim H7. rewrite H5. auto. -elim H6. rewrite H5. auto. -assumption. - -intro. unfold Interfere in H8. unfold redirect in H8. -destruct (OTFacts.eq_dec (fst_ext x0) (snd_ext e)); change_rewrite. -destruct H5. elim H7. rewrite H5. auto. -elim H1. rewrite in_pref. destruct H3. exists x1. -assert (eq (x, snd_ext e, Some x1) (snd_ext x0, fst_ext x0, Some x1)) by Eq_eq. -rewrite H9. rewrite <-H3. rewrite edge_comm. change Regs.registers with Regs.t. rewrite <-(edge_eq x0). auto. -destruct (OTFacts.eq_dec (snd_ext x0) (snd_ext e)); change_rewrite. -destruct H5. -elim H1. rewrite in_pref. destruct H3. exists x1. -assert (eq (x, snd_ext e, Some x1) (fst_ext x0, snd_ext x0, Some x1)) by Eq_eq. -rewrite H9. rewrite <-H3. rewrite <-(edge_eq x0). auto. -elim H7. rewrite H5. auto. -generalize (In_merge_edge_inv _ _ _ H Haff H8). intro. -destruct H9. destruct H9. destruct H10. -unfold redirect in H10. destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)). -unfold weak_eq in H10; destruct H10; destruct H10; change_rewrite. -destruct H5. elim H6. rewrite H5. auto. -elim H0. rewrite in_pref. destruct H3. exists x2. -assert (eq (x, fst_ext e, Some x2) (snd_ext x0, fst_ext x0, Some x2)) by Eq_eq. -rewrite H13. rewrite edge_comm. rewrite <-H3. change Regs.registers with Regs.t. rewrite <-(edge_eq x0). auto. -destruct H5. -elim H0. rewrite in_pref. destruct H3. exists x2. -assert (eq (x, fst_ext e, Some x2) (fst_ext x0, snd_ext x0, Some x2)) by Eq_eq. -rewrite H13. rewrite <-H3. rewrite <-(edge_eq x0). auto. -elim H6. rewrite H5. auto. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)). -unfold weak_eq in H10; destruct H10; destruct H10; change_rewrite. -destruct H5. -elim H0. rewrite in_pref. destruct H3. exists x2. -assert (eq (x, fst_ext e, Some x2) (fst_ext x0, snd_ext x0, Some x2)) by Eq_eq. -rewrite H13. rewrite <-H3. rewrite <-(edge_eq x0). auto. -elim H6. rewrite H5. auto. -destruct H5. elim H6. rewrite H5. auto. -elim H0. rewrite in_pref. destruct H3. exists x2. -assert (eq (x, fst_ext e, Some x2) (snd_ext x0, fst_ext x0, Some x2)) by Eq_eq. -rewrite H13. rewrite <-H3. rewrite edge_comm. change Regs.registers with Regs.t. rewrite <-(edge_eq x0). auto. -elim (interf_pref_conflict (fst_ext x0) (snd_ext x0) g). -unfold Prefere, Interfere. split. -destruct H3. exists x2. rewrite <-H3. change Regs.registers with Regs.t. rewrite <-(edge_eq x0). auto. -assert (eq (fst_ext x0, snd_ext x0, None) (fst_ext x1, snd_ext x1, None)). -unfold weak_eq in H10; destruct H10; destruct H10; change_rewrite. -Eq_eq. Eq_comm_eq. rewrite H12. -unfold same_type in H11. destruct H11; destruct H11. -destruct H13. simpl in H13. congruence. -rewrite <-H11. rewrite <-(edge_eq x1). auto. - -unfold redirect in H8. destruct (OTFacts.eq_dec (fst_ext x0) (snd_ext e)); change_rewrite. -destruct H5. elim H7. rewrite H5. auto. -elim H1. rewrite in_pref. destruct H3. exists x1. -assert (eq (x, snd_ext e, Some x1) (fst_ext x0, snd_ext x0, Some x1)) by Eq_comm_eq. -rewrite H9. rewrite <-H3. rewrite <-(edge_eq x0). auto. -destruct (OTFacts.eq_dec (snd_ext x0) (snd_ext e)); change_rewrite. -destruct H5. -elim H1. rewrite in_pref. destruct H3. exists x1. -assert (eq (x, snd_ext e, Some x1) (fst_ext x0, snd_ext x0, Some x1)) by Eq_eq. -rewrite H9. rewrite <-H3. rewrite <-(edge_eq x0). auto. -elim H7. rewrite H5. auto. -unfold Prefere in H8. destruct H8. -apply (move_related_charac2 _ _ (fst_ext x0, snd_ext x0, Some x1)). -unfold aff_edge. exists x1. auto. -assumption. -destruct H5;[left|right]; auto. -Qed. - -Lemma interfere_dec : forall x y g, -Interfere x y g \/ ~Interfere x y g. - -Proof. -intros. unfold Interfere. -destruct (RegRegProps.In_dec (x,y,None) (IE g)). -left. right. assumption. -right. intro. elim n. -rewrite In_graph_interf_edge_in_IE. split. -unfold interf_edge. auto. -assumption. -Qed. - -(* A vertex x different from r which is move-related in g and nonmove-related - in (merge e g p q) is either - 1) interfering with the first endpoint of e and prefering with the second one - 2) interfering with the second endpoint of e and prefering with the first one *) -Lemma move_merge_not_move : forall x e g p q, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -move_related (merge e g p q) x = false -> -move_related g x = true -> -(VertexSet.In x (VertexSet.inter (interference_adj (fst_ext e) g) (preference_adj (snd_ext e) g))) \/ -(VertexSet.In x (VertexSet.inter (interference_adj (snd_ext e) g) (preference_adj (fst_ext e) g))). - -Proof. -intros x e g p q H0 H1 H2 H3. generalize I. intro H. -generalize (move_related_charac _ _ H3). intro. -destruct H4. destruct H4. destruct H5. -cut (Interfere (fst_ext (redirect (snd_ext e) (fst_ext e) x0)) - (snd_ext (redirect (snd_ext e) (fst_ext e) x0)) - (merge e g p q)). intro. -unfold Interfere in H7. unfold redirect in H7. -destruct (OTFacts.eq_dec (fst_ext x0) (snd_ext e)); change_rewrite. -destruct H6. -elim H1. rewrite H6. auto. -generalize (In_merge_edge_inv e _ g p q H7). intro. -destruct H8. destruct H8. destruct H9 as [H9 HH9]. -assert (eq (fst_ext e, snd_ext x0, None) (redirect (snd_ext e) (fst_ext e) x1)). -unfold weak_eq; destruct H9; destruct H9; change_rewrite. -apply eq_ordered_eq. split; intuition. change_rewrite. rewrite redirect_weight_eq. -unfold same_type in HH9. destruct HH9. destruct H11. -destruct H12. simpl in H12. congruence. -destruct H11. rewrite <-H11. apply OptionN_as_OT.eq_refl. -Eq_comm_eq. simpl. rewrite redirect_weight_eq. -unfold same_type in HH9. destruct HH9. destruct H11. -destruct H12. simpl in H12. congruence. -destruct H11. rewrite <-H11. apply OptionN_as_OT.eq_refl. -generalize H10. clear H9 HH9 H10. intro H9. -unfold redirect in H9. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)). -destruct (eq_charac _ _ H9); change_rewrite; destruct H10. -elim (interf_pref_conflict x (snd_ext e) g). -split. -destruct H4. -unfold Prefere. exists x2. -cut (eq (x, snd_ext e, Some x2) x0). intro. -rewrite H12. auto. -rewrite (edge_eq x0). -rewrite edge_comm. apply eq_ordered_eq. constructor; simpl. -split; intuition. -rewrite H4. apply OptionN_as_OT.eq_refl. -unfold Interfere. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -rewrite (edge_eq x1). -rewrite edge_comm. apply eq_ordered_eq. constructor;simpl. -split; intuition. rewrite H6. auto. -generalize (get_weight_m _ _ (eq_sym H9)). simpl. intro. -rewrite H12. apply OptionN_as_OT.eq_refl. - -elim H0. rewrite H6. auto. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)). -destruct (eq_charac _ _ H9); change_rewrite. destruct H10. -elim H0. rewrite H6. auto. -destruct H10. - -elim (interf_pref_conflict x (snd_ext e) g). -split. -destruct H4. -unfold Prefere. exists x2. -cut (eq (x, snd_ext e, Some x2) x0). intro. -rewrite H12. auto. -rewrite (edge_eq x0). -rewrite edge_comm. apply eq_ordered_eq. constructor; simpl. -split; intuition. -rewrite H4. apply OptionN_as_OT.eq_refl. -unfold Interfere. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -rewrite (edge_eq x1). -apply eq_ordered_eq. constructor;simpl. -split; try rewrite H6; intuition. -generalize (get_weight_m _ _ (eq_sym H9)). simpl. intro. -rewrite H12. apply OptionN_as_OT.eq_refl. -left. apply VertexSet.inter_3. -rewrite in_interf. -cut (eq (x, fst_ext e, None) (fst_ext e, snd_ext x0, None)). intro. -rewrite H10. rewrite H9. auto. -rewrite edge_comm. apply eq_ordered_eq. -constructor; simpl. split; intuition. -apply OptionN_as_OT.eq_refl. -destruct H4. -rewrite in_pref. exists x2. -cut (eq (x,snd_ext e, Some x2) x0). intro. -rewrite H10. auto. -rewrite edge_comm. apply eq_ordered_eq. rewrite (edge_eq x0). -constructor; simpl. split; intuition. -auto. auto. rewrite H4. apply OptionN_as_OT.eq_refl. - -destruct (OTFacts.eq_dec (snd_ext x0) (snd_ext e)); change_rewrite. -destruct H6. -generalize (In_merge_edge_inv e _ g p q H7). intro. -destruct H8. destruct H8. destruct H9 as [H9 HH9]. -assert (eq (fst_ext x0, fst_ext e, None) (redirect (snd_ext e) (fst_ext e) x1)). -unfold weak_eq; destruct H9; destruct H9; change_rewrite. -Eq_eq. simpl. rewrite redirect_weight_eq. -unfold same_type in HH9. destruct HH9. destruct H11. -destruct H12. simpl in H12. congruence. -destruct H11. rewrite <-H11. apply OptionN_as_OT.eq_refl. -Eq_comm_eq. simpl. rewrite redirect_weight_eq. -unfold same_type in HH9. destruct HH9. destruct H11. -destruct H12. simpl in H12. congruence. -destruct H11. rewrite <-H11. apply OptionN_as_OT.eq_refl. -generalize H10. clear H9 HH9 H10. intro H9. -unfold redirect in H9. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)). -destruct (eq_charac _ _ H9); change_rewrite; destruct H10. -elim H0. rewrite H6. auto. -elim (interf_pref_conflict x (snd_ext e) g). -split. -destruct H4. -unfold Prefere. exists x2. -cut (eq (x, snd_ext e, Some x2) x0). intro. -rewrite H12. auto. -rewrite (edge_eq x0). -apply eq_ordered_eq. constructor; simpl. -split; intuition. -auto. -auto. -rewrite H4. apply OptionN_as_OT.eq_refl. -unfold Interfere. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -rewrite (edge_eq x1). -rewrite edge_comm. apply eq_ordered_eq. constructor;simpl. -split; try rewrite H6; intuition. -generalize (get_weight_m _ _ (eq_sym H9)). simpl. intro. -rewrite H12. apply OptionN_as_OT.eq_refl. - -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)). -destruct (eq_charac _ _ H9); change_rewrite. destruct H10. -elim (interf_pref_conflict x (snd_ext e) g). -split. -destruct H4. -unfold Prefere. exists x2. -cut (eq (x, snd_ext e, Some x2) x0). intro. -rewrite H12. auto. -rewrite (edge_eq x0). -apply eq_ordered_eq. constructor; simpl. -split; intuition. -rewrite H4. apply OptionN_as_OT.eq_refl. -unfold Interfere. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -rewrite (edge_eq x1). -apply eq_ordered_eq. constructor;simpl. -split; try rewrite H6; intuition. -generalize (get_weight_m _ _ (eq_sym H9)). simpl. intro. -rewrite H12. apply OptionN_as_OT.eq_refl. - -destruct H10. elim H0. rewrite H6. auto. - -left. apply VertexSet.inter_3. -rewrite in_interf. -cut (eq (x, fst_ext e, None) (fst_ext x0, fst_ext e, None)). intro. -rewrite H10. rewrite H9. auto. -apply eq_ordered_eq. -constructor; simpl. split; intuition. -apply OptionN_as_OT.eq_refl. -destruct H4. -rewrite in_pref. exists x2. -cut (eq (x,snd_ext e, Some x2) x0). intro. -rewrite H10. auto. -apply eq_ordered_eq. rewrite (edge_eq x0). -constructor; simpl. split; intuition. -rewrite H4. apply OptionN_as_OT.eq_refl. - -elim H1. rewrite H6. auto. -generalize (In_merge_edge_inv e _ g p q H7). intro. -destruct H8. destruct H8. destruct H9 as [H9 HH9]. -assert (eq (fst_ext x0, snd_ext x0, None) (redirect (snd_ext e) (fst_ext e) x1)). -unfold weak_eq; destruct H9; destruct H9; change_rewrite. -Eq_eq. simpl. rewrite redirect_weight_eq. -unfold same_type in HH9. destruct HH9. destruct H11. -destruct H12. simpl in H12. congruence. -destruct H11. rewrite <-H11. apply OptionN_as_OT.eq_refl. -Eq_comm_eq. simpl. rewrite redirect_weight_eq. -unfold same_type in HH9. destruct HH9. destruct H11. -destruct H12. simpl in H12. congruence. -destruct H11. rewrite <-H11. apply OptionN_as_OT.eq_refl. -generalize H10. clear H9 HH9 H10. intro H9. -unfold redirect in H9. -destruct (OTFacts.eq_dec (fst_ext x1) (snd_ext e)). -destruct (eq_charac _ _ H9); change_rewrite; destruct H10. -destruct H6. elim H0. rewrite H6. auto. -right. apply VertexSet.inter_3. -rewrite in_interf. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -rewrite edge_comm. apply eq_ordered_eq. -constructor; simpl. split; intuition. -auto. rewrite H6. auto. -generalize (get_weight_m _ _ H9). simpl. intro. -rewrite H12. unfold get_weight. apply OptionN_as_OT.eq_refl. -destruct H4. -rewrite in_pref. exists x2. -cut (eq (x, fst_ext e, Some x2) x0). intro. -rewrite H12. auto. -rewrite edge_comm. apply eq_ordered_eq. rewrite (edge_eq x0). -constructor; simpl. split; intuition. -auto. auto. rewrite H4. apply OptionN_as_OT.eq_refl. - -destruct H6. -right. apply VertexSet.inter_3. -rewrite in_interf. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -rewrite edge_comm. apply eq_ordered_eq. -constructor; simpl. split; intuition. -auto. rewrite H6. auto. -generalize (get_weight_m _ _ H9). simpl. intro. -rewrite H12. unfold get_weight. apply OptionN_as_OT.eq_refl. -destruct H4. -rewrite in_pref. exists x2. -cut (eq (x, fst_ext e, Some x2) x0). intro. -rewrite H12. auto. -apply eq_ordered_eq. rewrite (edge_eq x0). -constructor; simpl. split; intuition. -auto. auto. rewrite H4. apply OptionN_as_OT.eq_refl. - -elim H0. rewrite H6. auto. -destruct (OTFacts.eq_dec (snd_ext x1) (snd_ext e)). -destruct (eq_charac _ _ H9); change_rewrite; destruct H10. -destruct H6. -right. apply VertexSet.inter_3. -rewrite in_interf. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -apply eq_ordered_eq. -constructor; simpl. split; intuition. -rewrite H6. auto. auto. -generalize (get_weight_m _ _ H9). simpl. intro. -rewrite H12. unfold get_weight. apply OptionN_as_OT.eq_refl. -destruct H4. -rewrite in_pref. exists x2. -cut (eq (x, fst_ext e, Some x2) x0). intro. -rewrite H12. auto. -apply eq_ordered_eq. rewrite (edge_eq x0). -constructor; simpl. split; intuition. -auto. auto. rewrite H4. apply OptionN_as_OT.eq_refl. - -elim H0. rewrite H6. auto. - -destruct H6. -elim H0. rewrite H6. auto. -right. apply VertexSet.inter_3. -rewrite in_interf. -cut (eq (x, snd_ext e, None) x1). intro. -rewrite H12. auto. -apply eq_ordered_eq. -constructor; simpl. split; intuition. -rewrite H6. auto. auto. -generalize (get_weight_m _ _ H9). simpl. intro. -rewrite H12. unfold get_weight. apply OptionN_as_OT.eq_refl. -destruct H4. -rewrite in_pref. exists x2. -cut (eq (x, fst_ext e, Some x2) x0). intro. -rewrite H12. auto. -rewrite edge_comm. apply eq_ordered_eq. rewrite (edge_eq x0). -constructor; simpl. split; intuition. -auto. auto. rewrite H4. apply OptionN_as_OT.eq_refl. - -elim (interf_pref_conflict (fst_ext x0) (snd_ext x0) g). -split. -unfold Prefere. destruct H4. exists x2. -rewrite <-H4. change Regs.registers with Regs.t. rewrite <-(edge_eq x0). assumption. -unfold Interfere. rewrite H9. assumption. - -(* cut *) -destruct (interfere_dec (fst_ext (redirect (snd_ext e) (fst_ext e) x0)) - (snd_ext (redirect (snd_ext e) (fst_ext e) x0)) - (merge e g p q)). auto. -cut (~eq e x0). intro. -generalize (In_merge_pref_edge e x0 g p q H5 H8 H4 H7). intro. clear H7. -unfold Prefere in H9. destruct H9. unfold redirect in H7. -destruct (OTFacts.eq_dec (fst_ext x0) (snd_ext e)); change_rewrite. -destruct H6. -elim H1. rewrite H6. auto. -assert (move_related (merge e g p q) x = true). -apply move_related_charac2 with (e:= (fst_ext e, snd_ext x0, Some x1)). -exists x1. auto. -auto. -right. change_rewrite. auto. -rewrite H2 in H9. inversion H9. - -destruct (OTFacts.eq_dec (snd_ext x0) (snd_ext e)); change_rewrite. -destruct H6. -assert (move_related (merge e g p q) x = true). -apply move_related_charac2 with (e:= (fst_ext x0, fst_ext e, Some x1)). -exists x1. auto. -auto. -left. change_rewrite. auto. -rewrite H2 in H9. inversion H9. - -elim H1. rewrite H6. auto. - -assert (move_related (merge e g p q) x = true). -apply move_related_charac2 with (e:= (fst_ext x0, snd_ext x0, Some x1)). -exists x1. auto. -auto. -destruct H6;[left|right]; auto. -rewrite H2 in H9. inversion H9. - -intro. destruct (eq_charac _ _ H8); destruct H9. -destruct H6. -elim H0. rewrite H6. auto. -elim H1. rewrite H6. auto. -destruct H6. -elim H1. rewrite H6. auto. -elim H0. rewrite H6. auto. -Qed. - -(* A move-related vertex of g which does not interfere with an endpoint - of e and prefere with the other one is move-related in (merge e g p q) *) -Lemma move_related_merge_true : forall x e g p q, -move_related g x = true -> -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -~VertexSet.In x (VertexSet.inter (interference_adj (fst_ext e ) g) - (preference_adj (snd_ext e) g)) -> -~VertexSet.In x (VertexSet.inter (interference_adj (snd_ext e) g) - (preference_adj (fst_ext e) g)) -> -move_related (merge e g p q) x = true. - -Proof. -intros x e g p q H0 H1 H2 H3 H4. generalize I. intro H. -case_eq (move_related (merge e g p q) x); intros. -reflexivity. -generalize (move_merge_not_move _ _ _ p q H1 H2 H5 H0). intro. -destruct H6;[elim (H3 H6)|elim (H4 H6)]. -Qed. - -(* A vertex which interferes with an endpoint of e, preferes with - the other one and has a preference degree of one is - nonmove-related in (merge e g p q) *) -Lemma move_related_change_charac : forall x e g p q, -VertexSet.In x (VertexSet.union (VertexSet.inter (interference_adj (fst_ext e) g) - (preference_adj (snd_ext e) g)) - (VertexSet.inter (interference_adj (snd_ext e) g) - (preference_adj (fst_ext e) g))) -> -pref_degree g x = 1 -> -move_related (merge e g p q) x = false. - -Proof. -unfold pref_degree. intros x e g p q H1 H2. generalize I. intro H. -assert (move_related g x = true). -case_eq (move_related g x); auto. intro. -rewrite move_related_card in H0. inversion H0. -unfold pref_degree. congruence. -assert (~Register.eq x (fst_ext e)) as Hfst. -intro. rewrite H3 in H1. -destruct (VertexSet.union_1 H1). -elim (not_in_interf_self (fst_ext e) g). -apply (VertexSet.inter_1 H4). -elim (not_in_pref_self (fst_ext e) g). -apply (VertexSet.inter_2 H4). -assert (~Register.eq x (snd_ext e)) as Hsnd. -intro. rewrite H3 in H1. -destruct (VertexSet.union_1 H1). -elim (not_in_pref_self (snd_ext e) g). -apply (VertexSet.inter_2 H4). -elim (not_in_interf_self (snd_ext e) g). -apply (VertexSet.inter_1 H4). -destruct (VertexSet.union_1 H1). -assert (VertexSet.Equal (preference_adj x g) (VertexSet.singleton (snd_ext e))). -apply cardinal_1_singleton. -apply pref_adj_comm. apply (VertexSet.inter_2 H3). -assumption. - -assert (VertexSet.Equal (preference_adj x (merge e g p q)) VertexSet.empty). -split; intros. -assert (VertexSet.Subset (preference_adj x (merge e g p q)) - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (preference_adj x g)))). -apply preference_adj_merge; assumption. -destruct (Register.eq_dec a (fst_ext e)). -elim (interf_pref_conflict (fst_ext e) x (merge e g p q)). -split. -rewrite in_pref in H5. destruct H5. -unfold Prefere. exists x0. -assert (eq (fst_ext e, x, Some x0) (a, x, Some x0)) by Eq_eq. -rewrite H7. assumption. -unfold Interfere. -assert (eq (fst_ext e, x,None) (redirect (snd_ext e) (fst_ext e) (fst_ext e, x,None))). -apply eq_ordered_eq. -split; simpl; try split; try apply Regs.eq_refl. -unfold redirect; change_rewrite. -destruct (OTFacts.eq_dec (fst_ext e) (snd_ext e)). apply Regs.eq_refl. -destruct (OTFacts.eq_dec x (snd_ext e)). apply Regs.eq_refl. apply Regs.eq_refl. -unfold redirect; change_rewrite; simpl. -destruct (OTFacts.eq_dec (fst_ext e) (snd_ext e)). apply Regs.eq_refl. -destruct (OTFacts.eq_dec x (snd_ext e)). simpl. -elim (not_in_pref_self (snd_ext e) g). -rewrite r in H3. apply (VertexSet.inter_2 H3). apply Regs.eq_refl. -auto. rewrite redirect_weight_eq. simpl. apply OptionN_as_OT.eq_refl. -rewrite H7. apply In_merge_interf_edge. -rewrite <-in_interf. apply interf_adj_comm. apply (VertexSet.inter_1 H3). -unfold interf_edge. auto. - -assert (VertexSet.In a (VertexSet.add (fst_ext e) (VertexSet.remove (snd_ext e) (preference_adj x g)))). -apply H6. assumption. -assert (VertexSet.In a (VertexSet.remove (snd_ext e) (preference_adj x g))). -apply VertexSet.add_3 with (x:=fst_ext e); auto. -rewrite H4 in H8. -assert (~Register.eq a (snd_ext e)). -intro. elim (VertexSet.remove_1 (Register.eq_sym _ _ H9) H8). -generalize (VertexSet.remove_3 H8). intro. -generalize (VertexSet.singleton_1 H10). intro. elim H9. auto. -elim (VertexSet.empty_1 H5). -case_eq (move_related (merge e g p q) x); intros. -elim (move_related_not_empty_pref _ _ H6). assumption. -reflexivity. - -(* symetric part *) -assert (VertexSet.Equal (preference_adj x g) (VertexSet.singleton (fst_ext e))). -apply cardinal_1_singleton. -apply pref_adj_comm. apply (VertexSet.inter_2 H3). -assumption. - -assert (VertexSet.Equal (preference_adj x (merge e g p q)) VertexSet.empty). -split; intros. -assert (VertexSet.Subset (preference_adj x (merge e g p q)) - (VertexSet.add (fst_ext e) - (VertexSet.remove (snd_ext e) (preference_adj x g)))). -apply preference_adj_merge; assumption. -destruct (Register.eq_dec a (fst_ext e)). -elim (interf_pref_conflict (fst_ext e) x (merge e g p q)). -split. -rewrite in_pref in H5. destruct H5. -unfold Prefere. exists x0. -assert (eq (fst_ext e, x, Some x0) (a, x, Some x0)) by Eq_eq. -rewrite H7. assumption. -unfold Interfere. -assert (eq (fst_ext e, x,None) (redirect (snd_ext e) (fst_ext e) (snd_ext e, x,None))). -apply eq_ordered_eq. -split; simpl; try split; auto. -unfold redirect; change_rewrite. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). apply Regs.eq_refl. -elim n. apply Regs.eq_refl. -unfold redirect; change_rewrite. -destruct (OTFacts.eq_dec (snd_ext e) (snd_ext e)). apply Regs.eq_refl. -elim n. apply Regs.eq_refl. -rewrite redirect_weight_eq. simpl. apply OptionN_as_OT.eq_refl. -rewrite H7. apply In_merge_interf_edge. -rewrite <-in_interf. apply interf_adj_comm. apply (VertexSet.inter_1 H3). -unfold interf_edge. auto. - -assert (VertexSet.In a (VertexSet.add (fst_ext e) (VertexSet.remove (snd_ext e) (preference_adj x g)))). -apply H6. assumption. -assert (VertexSet.In a (VertexSet.remove (snd_ext e) (preference_adj x g))). -apply VertexSet.add_3 with (x:=fst_ext e); auto. -rewrite H4 in H8. -generalize (VertexSet.remove_3 H8). intro. -generalize (VertexSet.singleton_1 H9). intro. elim n. auto. -elim (VertexSet.empty_1 H5). -case_eq (move_related (merge e g p q) x); intros. -elim (move_related_not_empty_pref _ _ H6). assumption. -reflexivity. -Qed. - -(* A move-related vertex of g which is not move-related in - (merge e g p q) has a preference degree of 1 *) -Lemma move_related_dec_1 : forall x e g p q, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -move_related g x = true -> -move_related (merge e g p q) x = false -> -pref_degree g x = 1. - -Proof. -unfold pref_degree. intros x e g p q H0 H1 H2 H3. generalize I. intro H. -generalize (preference_adj_merge_2 x e g p q H0 H1). intro. -cut (VertexSet.cardinal (preference_adj x g) <= 1). intro. -cut (VertexSet.cardinal (preference_adj x g) > 0). intro. -intuition. -generalize (move_related_not_empty_pref x g H2). intro. -case_eq (VertexSet.cardinal (preference_adj x g)); intros. -elim H6. apply empty_is_empty_1. apply (cardinal_inv_1 H7). -intuition. -generalize (not_move_related_empty_pref x (merge e g p q) H3). intro. -generalize (move_merge_not_move _ _ _ p q H0 H1 H3 H2). intro. -destruct H6. -assert (VertexSet.cardinal (VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) (preference_adj x g))) <= - (VertexSet.cardinal (preference_adj x (merge e g p q)))). -apply subset_cardinal. assumption. -rewrite H5 in H7. rewrite empty_cardinal in H7. -rewrite remove_cardinal_2 in H7. -rewrite <-remove_cardinal_1 with (x:=snd_ext e). -apply le_n_S. assumption. -apply pref_adj_comm. apply (VertexSet.inter_2 H6). -intro. generalize (VertexSet.remove_3 H8). intro. -elim (interf_pref_conflict (fst_ext e) x g). -split. -rewrite in_pref in H9. destruct H9. -unfold Prefere. exists x0. assumption. -unfold Interfere. -rewrite edge_comm. generalize (VertexSet.inter_1 H6). intro. -rewrite in_interf in H10. assumption. -assert (VertexSet.cardinal (VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) (preference_adj x g))) <= - (VertexSet.cardinal (preference_adj x (merge e g p q)))). -apply subset_cardinal. assumption. -rewrite H5 in H7. rewrite empty_cardinal in H7. -rewrite <-remove_cardinal_2 with (x:=snd_ext e). -rewrite <-remove_cardinal_1 with (x:=fst_ext e). -apply le_n_S. assumption. -apply VertexSet.remove_2. -apply (In_graph_edge_diff_ext e g p). -apply pref_adj_comm. apply (VertexSet.inter_2 H6). -intro. elim (interf_pref_conflict (snd_ext e) x g). -split. -rewrite in_pref in H8. destruct H8. -unfold Prefere. exists x0. assumption. -generalize (VertexSet.inter_1 H6). intro. -unfold Interfere. rewrite edge_comm. rewrite <-in_interf. assumption. -Qed. - -(* Again, meaningful theorem *) -Theorem Merge_move_evolution : forall x e g p q, -~Register.eq x (fst_ext e) -> -~Register.eq x (snd_ext e) -> -((move_related g x = true /\ move_related (merge e g p q) x = false) - <-> - (pref_degree g x = 1 /\ - (VertexSet.In x (VertexSet.inter (interference_adj (fst_ext e) g) (preference_adj (snd_ext e) g)) \/ - VertexSet.In x (VertexSet.inter (interference_adj (snd_ext e) g) (preference_adj (fst_ext e) g))))). - -Proof. -split; intros. -destruct H1. -split. apply (move_related_dec_1 x e g p q); auto. - apply (move_merge_not_move x e g p q); auto. -destruct H1. -split. case_eq (move_related g x); auto. intro. rewrite move_related_card in H3; congruence. - apply move_related_change_charac; auto. - destruct H2;[apply VertexSet.union_2|apply VertexSet.union_3]; auto. -Qed. diff --git a/backend/Merge_WL.v b/backend/Merge_WL.v deleted file mode 100755 index d9b27e9..0000000 --- a/backend/Merge_WL.v +++ /dev/null @@ -1,818 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Simplify_WL. -Require Import IRC_graph. -Require Import WS. -Require Import Edges. -Require Import Affinity_relation. -Require Import Interference_adjacency. -Require Import Remove_Vertex_WL. -Require Import Merge_Degree. - -Import Edge RegFacts Props OTFacts. - -Definition classify x g' palette wl := -if is_precolored x g' then wl else -let a := get_spillWL wl in -let b := get_freezeWL wl in -let c := get_simplifyWL wl in -let d := get_movesWL wl in -if has_low_degree g' palette x then - if move_related g' x then (a, VertexSet.add x b, c, d) - else (a, VertexSet.remove x b, VertexSet.add x c, d) -else (VertexSet.add x a, VertexSet.remove x b, c, d). - -Definition merge_wl3 e ircg g' (p : In_graph_edge e (irc_g ircg)) (q : aff_edge e) := -let wl := (irc_wl ircg) in -let g := (irc_g ircg) in -let palette := (pal ircg) in -let k := (irc_k ircg) in -let simplifyWL := get_simplifyWL wl in -let freezeWL := get_freezeWL wl in -let spillWL := get_spillWL wl in -let movesWL := get_movesWL wl in -let pre := precolored g in -let iadj_fst_ := interference_adj (fst_ext e) g in -let iadj_fst := VertexSet.diff iadj_fst_ pre in -let iadj_snd_ := interference_adj (snd_ext e) g in -let iadj_snd := VertexSet.diff iadj_snd_ pre in -let padj_fst_ := preference_adj (fst_ext e) g in -let padj_fst := VertexSet.diff padj_fst_ pre in -let padj_snd_ := preference_adj (snd_ext e) g in -let padj_snd := VertexSet.diff padj_snd_ pre in -let iadj_interf := VertexSet.inter iadj_fst iadj_snd in -let iadj_padj_interf := VertexSet.union - (VertexSet.inter padj_fst iadj_snd) - (VertexSet.inter padj_snd iadj_fst) in -let N := VertexSet.filter - (fun x => eq_K k (VertexSet.cardinal (interference_adj x g))) - iadj_interf in -let (mr, nmr) := VertexSet.partition (move_related g) N in -let M := VertexSet.filter - (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g k x) - iadj_padj_interf in -let simplifyWL_ := VertexSet.union simplifyWL nmr in -let simplifyWL' := VertexSet.union simplifyWL_ M in -let freezeWL__ := VertexSet.diff freezeWL M in -let freezeWL_ := VertexSet.union freezeWL__ mr in -let freezeWL' := VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) freezeWL_) in -let spillWL_ := VertexSet.diff spillWL N in -let spillWL' := VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) spillWL_) in -let movesWL' := AE_merge_up (preference_adj (fst_ext e) g') - padj_fst_ padj_snd_ e movesWL in -classify (fst_ext e) g' k (spillWL', freezeWL', simplifyWL', movesWL'). - - -Lemma ws_merge3 : forall e ircg H H0, -WS_properties (merge e (irc_g ircg) H H0) - (VertexSet.cardinal (pal ircg)) - (merge_wl3 e ircg (merge e (irc_g ircg) H H0) H H0). - -Proof. -intros e ircg HH HH0. -unfold merge_wl3. unfold WS_properties. rewrite <-(Hk ircg) in *. -set (wl := irc_wl ircg) in *. -set (g := irc_g ircg) in *. -set (g' := merge e g HH HH0) in *. -set (palette := (pal ircg)) in *. -set (simplify := get_simplifyWL wl) in *. -set (freeze := get_freezeWL wl) in *. -set (spill := get_spillWL wl) in *. -set (moves := get_movesWL wl) in *. -set (k := VertexSet.cardinal palette) in *. -set (pre := precolored g) in *. -set (iadj_fst_ := interference_adj (fst_ext e) g) in *. -set (iadj_snd_ := interference_adj (snd_ext e) g) in *. -set (padj_fst_ := preference_adj (fst_ext e) g) in *. -set (padj_snd_ := preference_adj (snd_ext e) g) in *. -set (iadj_fst := VertexSet.diff iadj_fst_ pre) in *. -set (iadj_snd := VertexSet.diff iadj_snd_ pre) in *. -set (padj_fst := VertexSet.diff padj_fst_ pre) in *. -set (padj_snd := VertexSet.diff padj_snd_ pre) in *. -set (inter_interf := VertexSet.inter iadj_fst iadj_snd) in *. -set (iadj_padj_interf := VertexSet.union - (VertexSet.inter padj_fst iadj_snd) - (VertexSet.inter padj_snd iadj_fst)) in *. -set (N := VertexSet.filter - (fun x => eq_K k (VertexSet.cardinal (interference_adj x g))) - inter_interf) in *. -set (M := VertexSet.filter - (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) - && has_low_degree g k x) - iadj_padj_interf) in *. -set (mrnmr := VertexSet.partition (move_related g) N) in *. -case_eq mrnmr. intros mr nmr Hmr. unfold mrnmr in Hmr. -set (simplifyWL_ := VertexSet.union simplify nmr) in *. -set (simplifyWL' := VertexSet.union simplifyWL_ M) in *. -set (freezeWL__ := VertexSet.diff freeze M) in *. -set (freezeWL_ := VertexSet.union freezeWL__ mr) in *. -set (freezeWL' := VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) freezeWL_)) in *. -set (spillWL_ := VertexSet.diff spill N) in *. -set (spillWL' := VertexSet.remove (fst_ext e) (VertexSet.remove (snd_ext e) spillWL_)) in *. -set (movesWL' := AE_merge_up (preference_adj (fst_ext e) g') padj_fst_ padj_snd_ e moves) in *. - -unfold get_spillWL. unfold get_freezeWL. unfold get_simplifyWL. unfold get_movesWL. simpl. -generalize (HWS_irc ircg). intro HWS. rewrite <-(Hk ircg) in HWS. -generalize HH0. intro Haffa. destruct Haffa as [x2 Haffa]. - -cut (VertexSet.Equal mr (VertexSet.filter (move_related g) N)). -intro HMR. -cut (VertexSet.Equal nmr (VertexSet.filter (fun x => negb (move_related g x)) N)). -intro HNMR. - -split. -intro x. -(* spillWL for x = fst_ext e *) -destruct (Register.eq_dec x (fst_ext e)); split; intros. -(* spillWL => for x = fst_ext e *) -unfold classify in H. -unfold get_simplifyWL, get_freezeWL, get_spillWL, get_movesWL in H. simpl in H. -(* fst_ext e is precolored => False *) -case_eq (is_precolored (fst_ext e) g'); intros; rewrite H0 in H. -simpl in H. -unfold spillWL' in H. generalize (VertexSet.remove_3 (VertexSet.remove_3 H)). intro. -unfold spillWL_ in H1. generalize (VertexSet.diff_1 H1). intro. -generalize (In_spill_props _ _ _ _ _ _ _ _ H2 (refl_equal _) HWS). intro. -elim (proj2 (proj2 H3)). -apply VertexSet.remove_3 with (x:=snd_ext e). -rewrite <-(precolored_merge e (irc_g ircg) HH HH0). rewrite precolored_equiv. split. -rewrite is_precolored_ext with (y := fst_ext e); eassumption. -rewrite In_merge_vertex. split. intuition. -intro. elim (In_graph_edge_diff_ext e g). assumption. -rewrite <-e0. rewrite <-H4. auto. -(* fst_ext e is not precolored *) -case_eq (has_low_degree g' k (fst_ext e)); intros; rewrite H1 in H. -(* fst_ext e is of low-degree in g' *) -case_eq (move_related g' (fst_ext e)); intros; rewrite H2 in H; -simpl in H; unfold spillWL' in H; elim (VertexSet.remove_1 (Register.eq_sym _ _ e0) H). -(* fst_ext e is of high-degree in g' *) -simpl in H. split. -rewrite (compat_bool_low _ _ _ _ e0). assumption. -split. unfold g'. rewrite In_merge_vertex. split. -rewrite e0. apply (proj1 (In_graph_edge_in_ext _ _ HH)). -intro. elim (In_graph_edge_diff_ext e g). assumption. -rewrite <-e0. rewrite <-H2. auto. -rewrite precolored_equiv. intro. destruct H2. -rewrite (is_precolored_ext _ _ g' e0) in H2. rewrite H0 in H2. inversion H2. -(* spillWL <= for x = fst_ext e *) -destruct H. destruct H0. -unfold classify, get_spillWL, get_simplifyWL, get_freezeWL, get_movesWL. simpl. -case_eq (is_precolored (fst_ext e) g'); intros; simpl. -(* fst_ext e is precolored => False *) -elim H1. rewrite precolored_equiv. split. -rewrite (is_precolored_ext _ _ g' e0). assumption. assumption. -(* fst_ext e is not precolored *) -case_eq (has_low_degree g' k (fst_ext e)); intros; simpl. -(* fst_ext e is of low-degree in g' *) -rewrite (compat_bool_low _ _ _ _ e0) in H. rewrite H in H3. inversion H3. -(* fst_ext e is of high-degree in g' *) -apply VertexSet.add_1. intuition. - -(* spillWL : x is different from fst_ext e *) -assert (VertexSet.In x spillWL'). -unfold classify, get_simplifyWL, get_movesWL, get_freezeWL, get_spillWL in H. simpl in H. -case_eq (is_precolored (fst_ext e) g'); simpl in H; intro; rewrite H0 in H. -assumption. -case_eq (has_low_degree g' k (fst_ext e)); intros; rewrite H1 in H; simpl in H. -case_eq (move_related g' (fst_ext e)); intros; rewrite H2 in H; simpl in H; assumption. -apply VertexSet.add_3 with (x:=fst_ext e). auto. assumption. -generalize H0. clear H H0. intro H. -unfold spillWL' in H. generalize (VertexSet.remove_3 H). clear H. intro H. -assert (~Register.eq x (snd_ext e)). -intro. elim (VertexSet.remove_1 (Register.eq_sym _ _ H0) H). -generalize (VertexSet.remove_3 H). clear H. intro H. -unfold spillWL_ in H. -generalize (VertexSet.diff_1 H). intro. -generalize (VertexSet.diff_2 H). clear H. intro H. -generalize (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS). intro. -destruct H2. destruct H3. fold palette in H2. fold k in H2. -split. -case_eq (has_low_degree g' k x); intros. -generalize (merge_dec_interf _ _ _ _ HH HH0 H2 H5 n H0). intro. -generalize (merge_dec_K _ _ _ _ HH HH0 n H0 H2 H5). intro. -destruct H6. elim H. unfold N. -apply VertexSet.filter_3. -apply eq_K_compat. -unfold inter_interf. apply VertexSet.inter_3. -apply VertexSet.diff_3; assumption. -apply VertexSet.diff_3; assumption. -apply eq_K_1. assumption. -reflexivity. -split. unfold g'. rewrite In_merge_vertex. split; assumption. -unfold g'. rewrite precolored_merge. -intro. elim H4. apply (VertexSet.remove_3 H5). -(* spillWL <= for x different from fst_ext e *) -destruct H. destruct H0. -assert (VertexSet.In x spillWL'). -unfold spillWL'. apply VertexSet.remove_2. auto. -apply VertexSet.remove_2. intro. -unfold g' in H0. rewrite In_merge_vertex in H0. destruct H0. elim H3. auto. -unfold spillWL_. apply VertexSet.diff_3. -WS_apply HWS. -split. -apply (merge_low_1 g e x k HH0 HH). assumption. -intro. rewrite H2 in H0. -unfold g' in H0. rewrite In_merge_vertex in H0. destruct H0. elim H3. auto. -assumption. -split. unfold g' in H0. rewrite In_merge_vertex in H0. intuition. -unfold g' in H1. rewrite precolored_merge in H1. -intro. elim H1. apply VertexSet.remove_2. intro. -rewrite <-H3 in H0. -unfold g' in H0. rewrite In_merge_vertex in H0. destruct H0. elim H4. auto. -assumption. -intro. unfold N in H2. -generalize (VertexSet.filter_1 (eq_K_compat k g) H2). intro. -generalize (VertexSet.filter_2 (eq_K_compat k g) H2). clear H2. intro. -generalize (eq_K_2 _ _ H2). clear H2. intro H2. -assert (has_low_degree g' k x = true). -apply merge_dec_low. auto. -generalize (VertexSet.inter_1 H3). intro. apply (VertexSet.diff_1 H4). -generalize (VertexSet.inter_2 H3). intro. apply (VertexSet.diff_1 H4). -rewrite H in H4. inversion H4. -unfold classify, get_simplifyWL, get_freezeWL, get_movesWL, get_spillWL. simpl. -case_eq (is_precolored (fst_ext e) g'); intros. -assumption. -case_eq (has_low_degree g' k (fst_ext e)); intros. -case_eq (move_related g' (fst_ext e)); intros. -assumption. -assumption. -simpl. apply VertexSet.add_2. assumption. - -(* freezeWL *) -split. -intro x. destruct (Register.eq_dec x (fst_ext e)). -(* x = fst_ext e *) -split; intro. -(* => *) -unfold classify, get_spillWL, get_simplifyWL, get_movesWL, get_freezeWL in H. simpl in H. -case_eq (is_precolored (fst_ext e) g'); intros; rewrite H0 in H; simpl in H. -unfold freezeWL' in H. elim (VertexSet.remove_1 (Register.eq_sym _ _ e0) H). -case_eq (has_low_degree g' k (fst_ext e)); intros; rewrite H1 in H; simpl in H. -split. -rewrite (compat_bool_low _ _ _ _ e0). assumption. -case_eq (move_related g' (fst_ext e)); intros; rewrite H2 in H; simpl in H. -split. -rewrite (compat_bool_move _ _ _ e0). assumption. -rewrite precolored_equiv. intro. destruct H3. -rewrite (is_precolored_ext _ _ g' e0) in H3. rewrite H0 in H3. inversion H3. -elim (VertexSet.remove_1 (Register.eq_sym _ _ e0) H). -elim (VertexSet.remove_1 (Register.eq_sym _ _ e0) H). -(* <= *) -destruct H. destruct H0. -unfold classify, get_simplifyWL, get_spillWL, get_freezeWL, get_movesWL. simpl. -case_eq (is_precolored (fst_ext e) g'); intros; simpl. -elim H1. rewrite precolored_equiv. split. -rewrite (is_precolored_ext _ _ g' e0). assumption. -apply move_related_in_graph. assumption. -case_eq (has_low_degree g' k (fst_ext e)); intros; simpl. -case_eq (move_related g' (fst_ext e)); intros; simpl. -apply VertexSet.add_1. intuition. -rewrite (compat_bool_move _ _ _ e0) in H0. rewrite H0 in H4. inversion H4. -rewrite (compat_bool_low _ _ _ _ e0) in H. rewrite H3 in H. inversion H. - -(* x <> fst_ext e => *) -split; intro. -assert (VertexSet.In x freezeWL'). -unfold classify, get_spillWL, get_simplifyWL, get_freezeWL, get_movesWL in H. simpl in H. -case_eq (is_precolored (fst_ext e) g'); intros; rewrite H0 in H. -assumption. -case_eq (has_low_degree g' k (fst_ext e)); intros; rewrite H1 in H. -case_eq (move_related g' (fst_ext e)); intros; rewrite H2 in H. -simpl in H. apply VertexSet.add_3 with (x:= fst_ext e). auto. assumption. -simpl in H. apply (VertexSet.remove_3 H). -simpl in H. apply (VertexSet.remove_3 H). -generalize H0. clear H H0. intro H. -unfold freezeWL' in H. -generalize (VertexSet.remove_3 H). clear H. intro. -assert (~Register.eq x (snd_ext e)). -intro. elim (VertexSet.remove_1 (Register.eq_sym _ _ H0) H). -generalize (VertexSet.remove_3 H). clear H. intro. -unfold freezeWL_ in H. -destruct (VertexSet.union_1 H). -unfold freezeWL__ in H1. -generalize (VertexSet.diff_1 H1). intro. -generalize (VertexSet.diff_2 H1). clear H1. intro. -unfold freeze in H2. -generalize (In_freeze_props _ _ _ _ _ _ _ _ H2 (refl_equal _) HWS). intro. -destruct H3. destruct H4. destruct H5. -fold palette in H3. fold k in H3. -split. -apply low_dec; assumption. -split. -case_eq (move_related g' x); intros. -reflexivity. -elim H1. unfold M. -apply VertexSet.filter_3. -apply compat_move_up. -Require Import Merge_Move. -Require Import Graph_Facts. -generalize (move_merge_not_move x e g HH HH0 n H0 H7 H4). intro. -destruct H8. -apply VertexSet.union_3. -apply VertexSet.inter_3. -apply VertexSet.diff_3. apply (VertexSet.inter_2 H8). assumption. -apply VertexSet.diff_3. apply (VertexSet.inter_1 H8). assumption. -apply VertexSet.union_2. -apply VertexSet.inter_3. -apply VertexSet.diff_3. apply (VertexSet.inter_2 H8). assumption. -apply VertexSet.diff_3. apply (VertexSet.inter_1 H8). assumption. -generalize (move_related_dec_1 x e g HH HH0 n H0 H4 H7). intro. -unfold pref_degree in H8. rewrite (eq_K_1 _ _ H8). simpl. assumption. -unfold g'. rewrite precolored_merge. intro. elim H6. -apply (VertexSet.remove_3 H7). - -(* x in mr *) -rewrite HMR in H1. -generalize (VertexSet.filter_1 (compat_bool_move _) H1). intro. -generalize (VertexSet.filter_2 (compat_bool_move _) H1). clear H1. intro H1. -unfold N in H2. -generalize (VertexSet.filter_1 (eq_K_compat _ _) H2). intro. -generalize (VertexSet.filter_2 (eq_K_compat _ _) H2). clear H2. intro. -generalize (eq_K_2 _ _ H2). clear H2. intro. -split. -apply merge_dec_low. auto. -apply (VertexSet.diff_1 (VertexSet.inter_1 H3)). -apply (VertexSet.diff_1 (VertexSet.inter_2 H3)). -split. -apply move_related_merge_true; auto. -intro. -generalize (VertexSet.inter_1 H4). intro. -generalize (VertexSet.inter_2 H4). clear H4. intro. -generalize (VertexSet.inter_1 H3). intro. -generalize (VertexSet.inter_2 H3). clear H3. intro. -generalize (VertexSet.diff_1 H3). clear H3. intro. -generalize (VertexSet.diff_1 H6). intro. -generalize (VertexSet.diff_2 H6). clear H6. intro. -elim (interf_pref_conflict x (snd_ext e) g). -split. -rewrite in_pref in H4. destruct H4. -unfold Prefere. exists x0. assumption. -unfold Interfere. rewrite <-in_interf. assumption. -intro. -generalize (VertexSet.inter_1 H4). intro. -generalize (VertexSet.inter_2 H4). clear H4. intro. -generalize (VertexSet.inter_1 H3). intro. -generalize (VertexSet.inter_2 H3). clear H3. intro. -generalize (VertexSet.diff_1 H3). clear H3. intro. -generalize (VertexSet.diff_1 H6). intro. -generalize (VertexSet.diff_2 H6). clear H6. intro. -elim (interf_pref_conflict x (fst_ext e) g). -split. -rewrite in_pref in H4. destruct H4. -unfold Prefere. exists x0. assumption. -unfold Interfere. unfold iadj_fst_ in H7. rewrite in_interf in H7. auto. -unfold g'. rewrite precolored_merge. -intro. elim (VertexSet.diff_2 (VertexSet.inter_1 H3)). -apply (VertexSet.remove_3 H4). - -(* <= *) -assert (VertexSet.In x freezeWL'). -destruct H. destruct H0. -unfold freezeWL'. -apply VertexSet.remove_2. auto. -apply VertexSet.remove_2. -intro. elim (not_in_merge_snd e g HH HH0). -apply move_related_in_graph. rewrite (compat_bool_move _ _ _ H2). assumption. -unfold freezeWL_. -case_eq (has_low_degree g k x); intros. -apply VertexSet.union_2. -unfold freezeWL__. -apply VertexSet.diff_3. -WS_apply HWS. -split. assumption. -split. -apply (move_related_merge_move_related g e x HH HH0). assumption. -unfold g' in H1. rewrite precolored_merge in H1. intro. elim H1. -apply VertexSet.remove_2. intro. elim (not_in_merge_snd _ _ HH HH0). -apply move_related_in_graph. rewrite (compat_bool_move _ _ _ H4). assumption. assumption. -intro. unfold M in H3. -generalize (VertexSet.filter_2 (compat_move_up _ _) H3). intro. -assert (move_related g' x = false). -apply move_related_change_charac. -generalize (VertexSet.filter_1 (compat_move_up _ _) H3). intro. -unfold iadj_padj_interf in H5. -destruct (VertexSet.union_1 H5). -apply VertexSet.union_3. -apply VertexSet.inter_3. -apply (VertexSet.diff_1 (VertexSet.inter_2 H6)). -apply (VertexSet.diff_1 (VertexSet.inter_1 H6)). -apply VertexSet.union_2. -apply VertexSet.inter_3. -apply (VertexSet.diff_1 (VertexSet.inter_2 H6)). -apply (VertexSet.diff_1 (VertexSet.inter_1 H6)). -case_eq (eq_K 1 (VertexSet.cardinal (preference_adj x g))); intros. -symmetry. apply (eq_K_2 _ _ H5). rewrite H5 in H4. inversion H4. -rewrite H0 in H5. inversion H5. -apply VertexSet.union_3. -rewrite HMR. apply VertexSet.filter_3. -apply compat_bool_move. -unfold N. apply VertexSet.filter_3. -apply eq_K_compat. -generalize (merge_dec_interf x e k g HH HH0 H2 H n). intro. -destruct H3. -intro. elim (not_in_merge_snd e g HH HH0). apply move_related_in_graph. -rewrite (compat_bool_move _ _ _ (Register.eq_sym _ _ H3)). assumption. -apply VertexSet.inter_3. -apply VertexSet.diff_3. assumption. unfold pre. -unfold g' in H1. rewrite precolored_merge in H1. intro. elim H1. -apply VertexSet.remove_2. intro. -elim (not_in_merge_snd _ _ HH HH0). rewrite H6. apply move_related_in_graph. assumption. -assumption. -apply VertexSet.diff_3. assumption. unfold pre. -unfold g' in H1. rewrite precolored_merge in H1. intro. elim H1. -apply VertexSet.remove_2. intro. -elim (not_in_merge_snd _ _ HH HH0). rewrite H6. apply move_related_in_graph. assumption. -assumption. -apply eq_K_1. apply (merge_dec_K x e k g HH HH0); auto. -intro. elim (not_in_merge_snd e g HH HH0). -apply move_related_in_graph. rewrite (compat_bool_move _ _ _ H3) in H0. assumption. -apply (move_related_merge_move_related g e x HH HH0). assumption. - -unfold classify, get_spillWL, get_freezeWL, get_simplifyWL, get_movesWL; simpl. -case_eq (is_precolored (fst_ext e) g'); intro; simpl. -assumption. -case_eq (has_low_degree g' k (fst_ext e)); intro; simpl. -case_eq (move_related g' (fst_ext e)); intro; simpl. -apply VertexSet.add_2. assumption. -apply VertexSet.remove_2. auto. assumption. -apply VertexSet.remove_2. auto. assumption. - -(* simplifyWL *) -split. intro x. -(* x = fst_ext e *) -destruct (Register.eq_dec x (fst_ext e)). -(* => *) -split; intros. -unfold classify, get_spillWL, get_simplifyWL, get_movesWL, get_freezeWL in H. simpl in H. -case_eq (is_precolored (fst_ext e) g'); intros; rewrite H0 in H. -simpl in H. unfold simplifyWL' in H. -destruct (VertexSet.union_1 H). -unfold simplifyWL_ in H1. -destruct (VertexSet.union_1 H1). -generalize (In_simplify_props _ _ _ _ _ _ _ _ H2 (refl_equal _) HWS). intro. -destruct H3. destruct H4. destruct H5. -elim H6. apply VertexSet.remove_3 with (x:=snd_ext e). -rewrite <-(precolored_merge e (irc_g ircg) HH HH0). rewrite precolored_equiv. -rewrite (is_precolored_ext _ _ (merge e (irc_g ircg) HH HH0) e0). split. eassumption. -rewrite In_merge_vertex. split. assumption. -intro. elim (In_graph_edge_diff_ext e g). assumption. -rewrite <-H7. assumption. -rewrite HNMR in H2. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_move _)) H2). intro. -unfold N in H3. -generalize (VertexSet.filter_1 (eq_K_compat _ _) H3). intro. -generalize (VertexSet.inter_1 H4). intro. -generalize (VertexSet.inter_2 H4). clear H4. intro. -elim (VertexSet.diff_2 H5). unfold pre. -apply VertexSet.remove_3 with (x:=snd_ext e). -rewrite <-(precolored_merge e g HH HH0). rewrite precolored_equiv. -split. rewrite (is_precolored_ext _ _ (merge e g HH HH0) e0). eassumption. -rewrite In_merge_vertex. split. -rewrite e0. apply (proj1 (In_graph_edge_in_ext _ _ HH)). -intro. elim (not_in_interf_self (snd_ext e) g). rewrite H6 in H4. -apply (VertexSet.diff_1 H4). -unfold M in H1. -generalize (VertexSet.filter_1 (compat_move_up _ _) H1). intro. -unfold iadj_padj_interf in H2. -destruct (VertexSet.union_1 H2). -generalize (VertexSet.inter_1 H3). intro. generalize (VertexSet.diff_2 H4). intro. -elim H5. unfold pre. -apply VertexSet.remove_3 with (x:=snd_ext e). rewrite <-(precolored_merge e g HH HH0). -rewrite precolored_equiv. split. -rewrite (is_precolored_ext _ _ (merge e g HH HH0) e0). eassumption. -rewrite e0. rewrite In_merge_vertex. split. -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -intro. elim (In_graph_edge_diff_ext e g HH). auto. -generalize (VertexSet.inter_1 H3). intro. generalize (VertexSet.diff_2 H4). intro. -elim H5. unfold pre. -apply VertexSet.remove_3 with (x:=snd_ext e). rewrite <-(precolored_merge e g HH HH0). -rewrite precolored_equiv. split. -rewrite (is_precolored_ext _ _ (merge e g HH HH0) e0). eassumption. -rewrite e0. rewrite In_merge_vertex. split. -apply (proj1 (In_graph_edge_in_ext _ _ HH)). -intro. elim (In_graph_edge_diff_ext e g HH). auto. - -(* fst_ext e is not precolored *) -case_eq (has_low_degree g' k (fst_ext e)); intros; rewrite H1 in H. -case_eq (move_related g' (fst_ext e)); intros; rewrite H2 in H. -simpl in H. unfold simplifyWL' in H. -destruct (VertexSet.union_1 H). -unfold simplifyWL_ in H3. -destruct (VertexSet.union_1 H3). -assert (move_related g x = false). -apply (proj1 (proj2 (In_simplify_props _ _ _ _ _ _ _ _ H4 (refl_equal _) HWS))). -assert (move_related g x = true). -rewrite (compat_bool_move _ _ _ e0). -apply (proj1 (Aff_edge_aff _ _ HH HH0)). -rewrite H5 in H6. inversion H6. -rewrite HNMR in H4. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_move _)) H4). intro. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_move _)) H4). clear H4. intro H4. -unfold N in H5. -generalize (VertexSet.filter_1 (eq_K_compat _ _) H5). intro. -generalize (VertexSet.filter_2 (eq_K_compat _ _) H5). clear H5. intro. -generalize (eq_K_2 _ _ H5). clear H5. intro. -assert (move_related g x = true). -rewrite (compat_bool_move _ _ _ e0). -apply (proj1 (Aff_edge_aff _ _ HH HH0)). -rewrite H7 in H4. inversion H4. -unfold M in H3. -generalize (VertexSet.filter_1 (compat_move_up _ _) H3). intro. -unfold iadj_padj_interf in H4. -destruct (VertexSet.union_1 H4). -elim (not_in_pref_self (fst_ext e) g). rewrite e0 in H5. -apply (VertexSet.diff_1 (VertexSet.inter_1 H5)). -elim (not_in_interf_self (fst_ext e) g). rewrite e0 in H5. -apply (VertexSet.diff_1 (VertexSet.inter_2 H5)). -simpl in H. -split. rewrite (compat_bool_low _ _ _ _ e0). assumption. -split. rewrite (compat_bool_move _ _ _ e0). assumption. -split. unfold g'. rewrite In_merge_vertex. split. -rewrite e0. apply (proj1 (In_graph_edge_in_ext e g HH)). -intro. elim (In_graph_edge_diff_ext e g HH). rewrite <-H3. assumption. - -rewrite precolored_equiv. intro. destruct H3. -rewrite (is_precolored_ext _ _ g' e0) in H3. rewrite H0 in H3. inversion H3. -simpl in H. -unfold simplifyWL' in H. -destruct (VertexSet.union_1 H). -unfold simplifyWL_ in H2. -destruct (VertexSet.union_1 H2). -generalize (In_simplify_props _ _ _ _ _ _ _ _ H3 (refl_equal _) HWS). intro. -destruct H4. destruct H5. destruct H6. -generalize (proj1 (Aff_edge_aff _ _ HH HH0)). intro. -rewrite (compat_bool_move _ _ _ e0) in H5. fold g in H5. rewrite H5 in H8. inversion H8. -rewrite HNMR in H3. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_move _)) H3). intro. -generalize (proj1 (Aff_edge_aff _ _ HH HH0)). intro. -rewrite (compat_bool_move _ _ _ e0) in H4. rewrite H5 in H4. inversion H4. -unfold M in H2. -generalize (VertexSet.filter_1 (compat_move_up _ _) H2). intro. -unfold iadj_padj_interf in H3. -destruct (VertexSet.union_1 H3). -elim (not_in_pref_self (fst_ext e) g). rewrite e0 in H4. -apply (VertexSet.diff_1 (VertexSet.inter_1 H4)). -elim (not_in_interf_self (fst_ext e) g). rewrite e0 in H4. -apply (VertexSet.diff_1 (VertexSet.inter_2 H4)). - -(* <= *) -destruct H. destruct H0. destruct H1. -unfold classify, get_spillWL, get_simplifyWL, get_freezeWL, get_movesWL. simpl. -case_eq (is_precolored (fst_ext e) g'); intros; simpl. -rewrite precolored_equiv in H2. elim H2. -split. rewrite (is_precolored_ext _ _ g' e0). assumption. assumption. -case_eq (has_low_degree g' k (fst_ext e)); intros; simpl. -case_eq (move_related g' (fst_ext e)); intros; simpl. -rewrite (compat_bool_move _ _ _ e0) in H0. rewrite H0 in H5. inversion H5. -apply VertexSet.add_1. intuition. -rewrite (compat_bool_low _ _ _ _ e0) in H. rewrite H in H4. inversion H4. - -(* x <> fst_ext e *) -(* simplifyWL => *) -split; intros. -assert (VertexSet.In x simplifyWL'). -unfold classify, get_spillWL, get_simplifyWL, get_movesWL, get_freezeWL in H. simpl in H. -case_eq (is_precolored (fst_ext e) g'); intros; rewrite H0 in H. -assumption. -case_eq (has_low_degree g' k (fst_ext e)); intros; rewrite H1 in H. -case_eq (move_related g' (fst_ext e)); intros; rewrite H2 in H. -assumption. -simpl in H. apply VertexSet.add_3 with (x:=fst_ext e). auto. assumption. -assumption. -generalize H0. clear H H0. intro H. -unfold simplifyWL' in H. -destruct (VertexSet.union_1 H). -unfold simplifyWL_ in H0. -destruct (VertexSet.union_1 H0). -generalize (In_simplify_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS). intro. -destruct H2. destruct H3. destruct H4. -split. -apply low_dec; auto. -intro. generalize (proj2 (Aff_edge_aff e g HH HH0)). intro. -rewrite (compat_bool_move _ _ _ H6) in H3. fold g in H3. -rewrite H3 in H7. inversion H7. -split. -apply move_merge_false; auto. -split. unfold g'. rewrite In_merge_vertex. split. assumption. -intro. generalize (proj2 (Aff_edge_aff e g HH HH0)). intro. -rewrite (compat_bool_move _ _ _ H6) in H3. fold g in H3. -rewrite H3 in H7. inversion H7. -unfold g'. rewrite precolored_merge. intro. elim H5. -apply (VertexSet.remove_3 H6). -rewrite HNMR in H1. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_move _)) H1). intro. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_move _)) H1). clear H1. intro. -unfold N in H2. -generalize (VertexSet.filter_1 (eq_K_compat _ _) H2). intro. -generalize (VertexSet.filter_2 (eq_K_compat _ _) H2). clear H2. intro. -generalize (eq_K_2 _ _ H2). clear H2. intro. -split. -apply merge_dec_low. auto. -apply (VertexSet.diff_1 (VertexSet.inter_1 H3)). -apply (VertexSet.diff_1 (VertexSet.inter_2 H3)). -split. -apply move_merge_false; auto. case_eq (move_related g x); intros. -rewrite H4 in H1. inversion H1. reflexivity. -split. unfold g'. rewrite In_merge_vertex. split. -apply (in_interf_in x (fst_ext e) g). -apply (VertexSet.diff_1 (VertexSet.inter_1 H3)). -intro. generalize (proj2 (Aff_edge_aff e g HH HH0)). intro. -rewrite (compat_bool_move _ _ _ (Register.eq_sym _ _ H4)) in H5. -rewrite H5 in H1. inversion H1. -unfold g'. rewrite precolored_merge. intro. -elim (VertexSet.diff_2 (VertexSet.inter_1 H3)). apply (VertexSet.remove_3 H4). -unfold M in H0. -generalize (VertexSet.filter_1 (compat_move_up _ _) H0). intro. -generalize (VertexSet.filter_2 (compat_move_up _ _) H0). clear H0. intro. -assert (VertexSet.cardinal (preference_adj x g) = 1). -case_eq (eq_K 1 (VertexSet.cardinal (preference_adj x g))); intros. -generalize (eq_K_2 _ _ H2). clear H2. intro. auto. -rewrite H2 in H0. inversion H0. -generalize (eq_K_1 _ _ H2). intro. rewrite H3 in H0. simpl in H0. -split. -apply low_dec; auto. -intro. -destruct (VertexSet.union_1 H1). -elim (not_in_interf_self (snd_ext e) g). -rewrite H4 in H5. apply (VertexSet.diff_1 (VertexSet.inter_2 H5)). -elim (not_in_pref_self (snd_ext e) g). -rewrite H4 in H5. apply (VertexSet.diff_1 (VertexSet.inter_1 H5)). -split. -apply move_related_change_charac. -destruct (VertexSet.union_1 H1). -apply VertexSet.union_3. -apply VertexSet.inter_3. -apply (VertexSet.diff_1 (VertexSet.inter_2 H4)). -apply (VertexSet.diff_1 (VertexSet.inter_1 H4)). -apply VertexSet.union_2. -apply VertexSet.inter_3. -apply (VertexSet.diff_1 (VertexSet.inter_2 H4)). -apply (VertexSet.diff_1 (VertexSet.inter_1 H4)). -assumption. -split. -unfold g'. rewrite In_merge_vertex. split. -destruct (VertexSet.union_1 H1). -apply (in_interf_in x (snd_ext e) g). apply (VertexSet.diff_1 (VertexSet.inter_2 H4)). -apply (in_interf_in x (fst_ext e) g). apply (VertexSet.diff_1 (VertexSet.inter_2 H4)). -intro. destruct (VertexSet.union_1 H1). -elim (not_in_interf_self (snd_ext e) g). -rewrite H4 in H5. apply (VertexSet.diff_1 (VertexSet.inter_2 H5)). -elim (not_in_pref_self (snd_ext e) g). -rewrite H4 in H5. apply (VertexSet.diff_1 (VertexSet.inter_1 H5)). -unfold g'. rewrite precolored_merge. -intro. -destruct (VertexSet.union_1 H1). -elim (VertexSet.diff_2 (VertexSet.inter_1 H5)). apply (VertexSet.remove_3 H4). -elim (VertexSet.diff_2 (VertexSet.inter_1 H5)). apply (VertexSet.remove_3 H4). - -(* <= *) -destruct H. destruct H0. destruct H1. -assert (VertexSet.In x simplifyWL'). -unfold simplifyWL'. -case_eq (move_related g x); intros. -apply VertexSet.union_3. -unfold M. -apply VertexSet.filter_3. -apply compat_move_up. -assert (~Register.eq x (snd_ext e)) as Hsnd. -intro. elim (not_in_merge_snd e g HH HH0). rewrite H4 in H1. assumption. -generalize (move_merge_not_move _ _ _ _ HH0 n Hsnd H0 H3). intro. -destruct H4. -generalize (VertexSet.inter_1 H4). generalize (VertexSet.inter_2 H4). intros. -apply VertexSet.union_3. -apply VertexSet.inter_3. -apply VertexSet.diff_3. assumption. -unfold g' in H2. rewrite precolored_merge in H2. intro. elim H2. -apply VertexSet.remove_2. intro. elim (not_in_merge_snd _ _ HH HH0). -rewrite H8. assumption. -assumption. -apply VertexSet.diff_3. assumption. -unfold g' in H2. rewrite precolored_merge in H2. intro. elim H2. -apply VertexSet.remove_2. intro. elim (not_in_merge_snd _ _ HH HH0). -rewrite H8. assumption. -assumption. -generalize (VertexSet.inter_1 H4). generalize (VertexSet.inter_2 H4). intros. -apply VertexSet.union_2. -apply VertexSet.inter_3. -apply VertexSet.diff_3. assumption. -unfold g' in H2. -rewrite precolored_merge in H2. intro. elim H2. -apply VertexSet.remove_2. intro. elim (not_in_merge_snd _ _ HH HH0). -rewrite H8. assumption. -assumption. -apply VertexSet.diff_3. assumption. -unfold g' in H2. -rewrite precolored_merge in H2. intro. elim H2. -apply VertexSet.remove_2. intro. elim (not_in_merge_snd _ _ HH HH0). -rewrite H8. assumption. -assumption. - -assert (~Register.eq x (snd_ext e)) as Hsnd. -intro. rewrite H4 in H1. elim (not_in_merge_snd e g HH HH0 H1). -generalize (move_related_dec_1 _ _ _ HH HH0 n Hsnd H3 H0). intro. -unfold pref_degree in H4. rewrite (eq_K_1 _ _ H4). simpl. -case_eq (has_low_degree g k x); intros. -reflexivity. -generalize (move_merge_not_move _ _ _ _ HH0 n Hsnd H0 H3). intro. -assert (~Register.eq x (snd_ext e)). -intro. rewrite H7 in H1. elim (not_in_merge_snd e g HH HH0 H1). -generalize (merge_dec_interf _ _ _ _ HH HH0 H5 H n H7). intro. -destruct H8. destruct H6. -generalize (VertexSet.inter_2 H6). intro. -elim (interf_pref_conflict x (snd_ext e) g). -rewrite in_pref in H10. -split. -unfold Prefere. assumption. -unfold Interfere. rewrite <-in_interf. assumption. -elim (interf_pref_conflict x (fst_ext e) g). -generalize (VertexSet.inter_2 H6). intro. -rewrite in_pref in H10. -split. -unfold Prefere. assumption. -unfold Interfere. rewrite <-in_interf. auto. -apply VertexSet.union_2. -unfold simplifyWL_. -case_eq (has_low_degree g k x); intros. -apply VertexSet.union_2. - -WS_apply HWS. -split. assumption. -split. assumption. -split. unfold g' in H1. rewrite In_merge_vertex in H1. intuition. -unfold g' in H2. rewrite precolored_merge in H2. -intro. elim H2. apply VertexSet.remove_2. -intro. apply (not_in_merge_snd _ _ HH HH0). rewrite H6. assumption. -assumption. - -apply VertexSet.union_3. -rewrite HNMR. -apply VertexSet.filter_3. -apply compat_not_compat. apply compat_bool_move. -unfold N. apply VertexSet.filter_3. -apply eq_K_compat. -assert (~Register.eq x (snd_ext e)). -intro. rewrite H5 in H1. elim (not_in_merge_snd e g HH HH0 H1). -generalize (merge_dec_interf _ _ _ _ HH HH0 H4 H n H5). -intro. destruct H6. apply VertexSet.inter_3. -apply VertexSet.diff_3. assumption. -unfold g' in H2. rewrite precolored_merge in H2. -intro. elim H2. apply VertexSet.remove_2. auto. -assumption. -apply VertexSet.diff_3. assumption. -unfold g' in H2. rewrite precolored_merge in H2. -intro. elim H2. apply VertexSet.remove_2. auto. -assumption. -apply eq_K_1. -apply (merge_dec_K x e k g HH HH0 n). -intro. rewrite H5 in H1. elim (not_in_merge_snd e g HH HH0 H1). -assumption. -assumption. -rewrite H3. auto. - -unfold classify, get_spillWL, get_simplifyWL, get_movesWL, get_freezeWL; simpl. -case_eq (is_precolored (fst_ext e) g'); intros. -assumption. -case_eq (has_low_degree g' k (fst_ext e)); intros. -case_eq (move_related g' (fst_ext e)); intros. -assumption. -simpl. apply VertexSet.add_2; auto. -assumption. - -(* moves !!!!! *) -split; intros. -assert (EdgeSet.In e0 movesWL'). -unfold classify,get_spillWL, get_simplifyWL, get_movesWL, get_freezeWL in H; simpl in H; -destruct (is_precolored (fst_ext e) g'); -destruct (has_low_degree g' k (fst_ext e)); -destruct (move_related g' (fst_ext e)); simpl; assumption. -unfold movesWL' in H0. -unfold g'. rewrite <-AE_merge_wl. eassumption. -intros. rewrite <-In_graph_aff_edge_in_AE. -rewrite moves_AE. unfold moves. reflexivity. eassumption. - -destruct H. simpl. -assert (EdgeSet.In e0 movesWL'). -unfold movesWL'. unfold g', padj_fst_, padj_snd_. rewrite AE_merge_wl. -split; assumption. - -intros. rewrite <-In_graph_aff_edge_in_AE. -rewrite moves_AE. unfold moves. reflexivity. eassumption. -unfold classify,get_spillWL, get_simplifyWL, get_movesWL, get_freezeWL; simpl; -destruct (is_precolored (fst_ext e) g'); -destruct (has_low_degree g' k (fst_ext e)); -destruct (move_related g' (fst_ext e)); simpl; assumption. - -rewrite <-VertexSet.partition_2. rewrite Hmr. intuition. -apply compat_bool_move. -rewrite <-VertexSet.partition_1. rewrite Hmr. intuition. -apply compat_bool_move. -Qed. - -Lemma WS_coalesce : forall e ircg H H0, -WS_properties (merge e (irc_g ircg) H H0) - (irc_k ircg) - (merge_wl3 e ircg (merge e (irc_g ircg) H H0) H H0). - -Proof. -intros. rewrite <-(Hk ircg). apply ws_merge3. -Qed.
\ No newline at end of file diff --git a/backend/MyAllocation.v b/backend/MyAllocation.v deleted file mode 100644 index 0d5f3e3..0000000 --- a/backend/MyAllocation.v +++ /dev/null @@ -1,1434 +0,0 @@ -Require Import IRC. -Require Import IRCColoring. -Require Import Graph_translation. -Require Import MyRegisters. -Require Import Locations. -Require Import RTLtyping. -Require Import ZArith. -Require Import AST. -Require Import Typed_interfgraphs. -Require Import Edges. -Require Import Graph_Facts. -Require Import Interference_adjacency. -Require Import InterfGraph. -Require Import Conventions. -Require Import Palettes. -Require Import InterfGraph_Construction. -Require Import WS. -Require Import Conservative_criteria. -Require Import IRC_graph. -Require Import IRC_Graph_Functions. -Require Import InterfGraphMapImp. -Require Import Registers. - -Import Props Edge RegFacts. - -Module ColFacts := FMapFacts.Facts ColorMap. - -Definition graph_coloring_aux x int_map float_map env := -match (Regs.get_type (Regs.P x) env) with -| Tint => match (map_to_coloring int_map (Regs.P x)) with - | Some (Regs.P z) => S (Local Z0 Tint) - | Some (Regs.M z) => R z - | None => S (Local (Zpos x) Tint) - end -| Tfloat => match (map_to_coloring float_map (Regs.P x)) with - | Some (Regs.P z) => S (Local Z0 Tfloat) - | Some (Regs.M z) => R z - | None => S (Local (Zpos x) Tfloat) - end -end. - -Definition reg_translation s := -Regset.fold (fun v s => VertexSet.add (Regs.reg_to_Reg v) s) s VertexSet.empty. - -Definition mreg_translation s := -MRegset.fold (fun v s => VertexSet.add (Regs.mreg_to_Reg v) s) s VertexSet.empty. - -Definition Typed_interfgraphs g env := -let regreg_interf_partition := -regreg_edge_type_partition (interf_reg_reg g) env in -let int_regreg_interf_edge := rr1 regreg_interf_partition in -let float_regreg_interf_edge := rr2 regreg_interf_partition in -let int_regreg_interf_reg := rr3 regreg_interf_partition in -let float_regreg_interf_reg := rr4 regreg_interf_partition in -let regmreg_interf_partition := -regmreg_edge_type_partition (interf_reg_mreg g) env in -let int_regmreg_interf_edge := rm1 regmreg_interf_partition in -let float_regmreg_interf_edge := rm2 regmreg_interf_partition in -let int_regmreg_interf_reg := rm3 regmreg_interf_partition in -let float_regmreg_interf_reg := rm4 regmreg_interf_partition in -let int_regmreg_interf_mreg := rm5 regmreg_interf_partition in -let float_regmreg_interf_mreg := rm6 regmreg_interf_partition in -let regreg_pref_partition := -regreg_edge_type_partition (pref_reg_reg g) env in -let int_regreg_pref_edge := rr1 regreg_pref_partition in -let float_regreg_pref_edge := rr2 regreg_pref_partition in -let int_regreg_pref_reg := rr3 regreg_pref_partition in -let float_regreg_pref_reg := rr4 regreg_pref_partition in -let regmreg_pref_partition := -regmreg_edge_type_partition (pref_reg_mreg g) env in -let int_regmreg_pref_edge := rm1 regmreg_pref_partition in -let float_regmreg_pref_edge := rm2 regmreg_pref_partition in -let int_regmreg_pref_reg := rm3 regmreg_pref_partition in -let float_regmreg_pref_reg := rm4 regmreg_pref_partition in -let int_regmreg_pref_mreg := rm5 regmreg_pref_partition in -let float_regmreg_pref_mreg := rm6 regmreg_pref_partition in -let int_regs := Regset.union int_regreg_interf_reg - (Regset.union int_regmreg_interf_reg - (Regset.union int_regreg_pref_reg int_regmreg_pref_reg)) in -let float_regs := Regset.union float_regreg_interf_reg - (Regset.union float_regmreg_interf_reg - (Regset.union float_regreg_pref_reg float_regmreg_pref_reg)) in -let int_mregs := MRegset.union int_regmreg_interf_mreg int_regmreg_pref_mreg in -let float_mregs := MRegset.union float_regmreg_interf_mreg float_regmreg_pref_mreg in -let int_Regs := VertexSet.union (reg_translation int_regs) (mreg_translation int_mregs) in -let float_Regs := VertexSet.union (reg_translation float_regs) (mreg_translation float_mregs) in -(int_Regs, -mkgraph int_regreg_interf_edge int_regmreg_interf_edge int_regreg_pref_edge int_regmreg_pref_edge, -float_Regs, -mkgraph float_regreg_interf_edge float_regmreg_interf_edge float_regreg_pref_edge float_regmreg_pref_edge). - -Lemma extremities_int_interf_graph : forall g env, -forall e, EdgeSet.In e (interfgraph_affinity_edges (snd (fst (fst (Typed_interfgraphs g env))))) \/ - EdgeSet.In e (interfgraph_interference_edges (snd (fst (fst (Typed_interfgraphs g env))))) -> - VertexSet.In (fst_ext e) (fst (fst (fst (Typed_interfgraphs g env)))) /\ - VertexSet.In (snd_ext e) (fst (fst (fst (Typed_interfgraphs g env)))). - -Proof. -Admitted. - -Lemma extremities_float_interf_graph : forall g env, -forall e, EdgeSet.In e (interfgraph_affinity_edges (snd (Typed_interfgraphs g env))) \/ - EdgeSet.In e (interfgraph_interference_edges (snd (Typed_interfgraphs g env))) -> - VertexSet.In (fst_ext e) (snd (fst (Typed_interfgraphs g env))) /\ - VertexSet.In (snd_ext e) (snd (fst (Typed_interfgraphs g env))). - -Proof. -Admitted. - -Definition my_graph_coloring g env := -let typed_graphs := Typed_interfgraphs g env in -let intR := fst (fst (fst typed_graphs)) in -let intG := snd (fst (fst typed_graphs)) in -let floatR := snd (fst typed_graphs) in -let floatG := snd typed_graphs in -let int_graph := graph_translation intG intR (extremities_int_interf_graph g env) in -let float_graph := graph_translation floatG floatR (extremities_float_interf_graph g env) in -let int_map := (IRC_map (graph_to_IRC_graph int_graph int_palette)) in -let float_map := (IRC_map (graph_to_IRC_graph float_graph float_palette)) in -fun x => graph_coloring_aux x int_map float_map env. - -Section Coloring_to_allocation. - -Variable g : graph. -Variable env : regenv. -Definition typed_graphs := Typed_interfgraphs g env. -Definition intR := fst (fst (fst typed_graphs)). -Definition intG := snd (fst (fst typed_graphs)). -Definition floatR := snd (fst typed_graphs). -Definition floatG := snd typed_graphs. -Definition int_graph := graph_translation intG intR (extremities_int_interf_graph g env). -Definition float_graph := graph_translation floatG floatR (extremities_float_interf_graph g env). -Definition int_map := (IRC_map (graph_to_IRC_graph int_graph int_palette)). -Definition float_map := (IRC_map (graph_to_IRC_graph float_graph float_palette)). -Definition int_coloring := map_to_coloring int_map. -Definition float_coloring := map_to_coloring float_map. - -Hypothesis temporaries_out : forall x, -In_graph (Regs.M x) int_graph -> ~List.In (R x) temporaries. - -Hypothesis correct_palette_int : forall x, -VertexSet.In x (precolored int_graph) -> VertexSet.In x int_palette. - -Hypothesis correct_palette_float : forall x, -VertexSet.In x (precolored float_graph) -> VertexSet.In x float_palette. - -Lemma proper_coloring_int : proper_coloring int_coloring int_graph int_palette. - -Proof. -intros. apply proper_coloring_IRC_aux. -intro. apply correct_palette_int. -Qed. - -Lemma proper_coloring_float : proper_coloring float_coloring float_graph float_palette. - -Proof. -intros. apply proper_coloring_IRC_aux. -intro. apply correct_palette_float. -Qed. - -Import SetoidList. - -Lemma exists_refl : forall x, -exists y, Regs.M x = Regs.M y. - -Proof. -intro x. exists x. auto. -Qed. - -Lemma mreg_int_palette : forall x, -VertexSet.In x int_palette -> -exists y, x = Regs.M y. - -Proof. -unfold int_palette. intros x H. -repeat (destruct (proj1 (Props.Dec.F.add_iff _ _ _) H); - [inversion H0;subst; apply exists_refl|generalize H0; clear H H0; intro H]). -elim (VertexSet.empty_1 H). -Qed. - -Lemma mreg_float_palette : forall x, -VertexSet.In x float_palette -> -exists y, x = Regs.M y. - -Proof. -unfold float_palette. intros x H. -repeat (destruct (proj1 (Props.Dec.F.add_iff _ _ _) H); - [inversion H0;subst; apply exists_refl|generalize H0; clear H H0; intro H]). -elim (VertexSet.empty_1 H). -Qed. - -Lemma register_heuristic_mreg : forall x r, -(IRC int_graph int_palette) x = Some r -> -exists y, r = Regs.M y. - -Proof. -intros x r H. -apply mreg_int_palette. -generalize (proper_coloring_IRC_aux int_graph int_palette correct_palette_int). -intro H0. -unfold proper_coloring in H0. -unfold proper_coloring_3 in H0. -do 2 destruct H0 as [_ H0]. -apply H0 with (x := x). -rewrite H. apply OptionReg.eq_refl. -Qed. - -Lemma register_heuristic_mreg_float : forall x r, -(IRC float_graph float_palette) x = Some r -> -exists y, r = Regs.M y. - -Proof. -intros x r H. -apply mreg_float_palette. -generalize (proper_coloring_IRC_aux float_graph float_palette correct_palette_float). -intro H0. -unfold proper_coloring in H0. -unfold proper_coloring_3 in H0. -do 2 destruct H0 as [_ H0]. -apply H0 with (x := x). -rewrite H. apply OptionReg.eq_refl. -Qed. - -Lemma int_palette_type : forall x, -VertexSet.In x int_palette -> -Regs.get_type x env = Tint. - -Proof. -unfold int_palette. intros x H. -repeat (destruct (proj1 (Props.Dec.F.add_iff _ _ _) H); - [inversion H0;subst; auto|generalize H0; clear H H0; intro H]). -elim (VertexSet.empty_1 H). -Qed. - -Lemma float_palette_type : forall x, -VertexSet.In x float_palette -> -Regs.get_type x env = Tfloat. - -Proof. -unfold float_palette. intros x H. -repeat (destruct (proj1 (Props.Dec.F.add_iff _ _ _) H); - [inversion H0;subst; auto|generalize H0; clear H H0; intro H]). -elim (VertexSet.empty_1 H). -Qed. - -Lemma register_heuristic_type_int : forall x r, -IRC int_graph int_palette x = Some r -> -Regs.get_type r env = Tint. - -Proof. -intros x r H. -apply int_palette_type. -generalize (proper_coloring_IRC_aux int_graph int_palette (correct_palette_int)). -intro H0. -unfold proper_coloring in H0. do 2 destruct H0 as [_ H0]. -unfold proper_coloring_3 in H0. -apply (H0 x r). -rewrite H. apply OptionReg.eq_refl. -Qed. - -Lemma register_heuristic_type_float : forall x r, -IRC float_graph float_palette x = Some r -> -Regs.get_type r env = Tfloat. - -Proof. -intros x r H. -apply float_palette_type. -generalize (proper_coloring_IRC_aux float_graph float_palette correct_palette_float). -intro H0. -unfold proper_coloring in H0. do 2 destruct H0 as [_ H0]. -unfold proper_coloring_3 in H0. -apply (H0 x r). -rewrite H. apply OptionReg.eq_refl. -Qed. - -Lemma Loc_reg_eq_type : forall x, -Regs.get_type (Regs.P x) env = Loc.type (my_graph_coloring g env x). - -Proof. -intro x. -unfold my_graph_coloring. -change (snd (fst (fst (Typed_interfgraphs g env)))) with intG. -change (fst (fst (fst (Typed_interfgraphs g env)))) with intR. -change (snd (fst (Typed_interfgraphs g env))) with floatR. -change (snd (Typed_interfgraphs g env)) with floatG. -fold int_graph; fold float_graph. -unfold graph_coloring_aux. -case_eq (Regs.get_type (Regs.P x) env); intros HH. -change (map_to_coloring (IRC_map (graph_to_IRC_graph int_graph int_palette))) with - (IRC int_graph int_palette). -case_eq (IRC int_graph int_palette (Regs.P x)); intros. -generalize (register_heuristic_mreg _ _ H). intro. destruct H0. rewrite H0. -generalize (register_heuristic_type_int _ _ H). -unfold Regs.get_type. rewrite H0. simpl. auto. -simpl. auto. -change (map_to_coloring (IRC_map (graph_to_IRC_graph float_graph float_palette))) with - (IRC float_graph float_palette). -case_eq (IRC float_graph float_palette (Regs.P x)); intros. -generalize (register_heuristic_mreg_float _ _ H). intro. destruct H0. rewrite H0. -generalize (register_heuristic_type_float _ _ H). -unfold Regs.get_type. rewrite H0. simpl. auto. -simpl. auto. -Qed. - -Lemma regreg_in_fst_partition : forall e s, -SetRegReg.In e s -> -env (fst e) = Tint -> -env (snd e) = Tint -> -SetRegReg.In e (rr1 (regreg_edge_type_partition s env)). - -Proof. -intros. -unfold regreg_edge_type_partition. -set (f:=(fun (e0 : SetRegReg.elt) (s0 : regregpartition) => - match env (fst e0) with - | Tint => - match env (snd e0) with - | Tint => - (SetRegReg.add e0 (rr1 s0), rr2 s0, - Regset.add (fst e0) (Regset.add (snd e0) (rr3 s0)), - rr4 s0) - | Tfloat => - (rr1 s0, rr2 s0, Regset.add (fst e0) (rr3 s0), - Regset.add (snd e0) (rr4 s0)) - end - | Tfloat => - match env (snd e0) with - | Tint => - (rr1 s0, rr2 s0, Regset.add (snd e0) (rr3 s0), - Regset.add (fst e0) (rr4 s0)) - | Tfloat => - (rr1 s0, SetRegReg.add e0 (rr2 s0), rr3 s0, - Regset.add (fst e0) (Regset.add (snd e0) (rr4 s0))) - end - end)). -unfold regregpartition in *. fold f. - -generalize (SetRegReg.empty,SetRegReg.empty, Regset.empty, Regset.empty). -generalize SetRegReg.elements_1. intro HH. -generalize (HH s e H). clear H HH. intro HH. -intro p. rewrite SetRegReg.fold_1. generalize p. clear p. -induction (SetRegReg.elements s). -inversion HH. -inversion HH. - subst. intro p. do 3 destruct p. simpl. - -assert (f a (t2,t3,t1,t0) = (SetRegReg.add a t2, t3, Regset.add (fst a) (Regset.add (snd a) t1),t0)). -unfold f. - -destruct H2. rewrite H in *. rewrite H2 in *. rewrite H0. rewrite H1. simpl. reflexivity. -rewrite H. -destruct H2. - -assert (forall x s1 s2 s3 s4, SetRegReg.In x s1 -> - SetRegReg.In x (rr1 (fold_left - (fun (a0 : SetRegReg.t*SetRegReg.t*Regset.t*Regset.t) (e0 : SetRegReg.elt) => f e0 a0) - l (s1, s2, s3, s4)))). - -clear H H0 H1 H2 HH IHl. - -induction l. -simpl. auto. -intros x s1 s2 s3 s4 H2. -simpl. - -assert (f a0 (s1,s2,s3,s4) = (SetRegReg.add a0 s1, s2,Regset.add (fst a0) (Regset.add (snd a0) s3), s4) \/ - f a0 (s1,s2,s3,s4) = (s1, SetRegReg.add a0 s2, s3, Regset.add (fst a0)(Regset.add (snd a0) s4)) \/ - f a0 (s1,s2,s3,s4) = (s1,s2,Regset.add (fst a0) s3, Regset.add (snd a0) s4)\/ - f a0 (s1,s2,s3,s4) = (s1,s2,Regset.add (snd a0) s3, Regset.add (fst a0) s4)). - -unfold f. -destruct (env (fst a0)); destruct (env (snd a0)); -unfold rr1,rr2,rr3,rr4; simpl; auto. - -destruct H. - -rewrite H. apply IHl. apply SetRegReg.add_2. assumption. - -destruct H. -rewrite H. -apply IHl. assumption. -destruct H; rewrite H. -apply IHl. assumption. -apply IHl. assumption. - -apply H4. apply SetRegReg.add_1. intuition. - -subst. simpl. intro p. apply IHl. assumption. -Qed. - -Lemma regreg_in_snd_partition : forall e s, -SetRegReg.In e s -> -env (fst e) = Tfloat -> -env (snd e) = Tfloat -> -SetRegReg.In e (rr2 (regreg_edge_type_partition s env)). - -Proof. -intros. -unfold regreg_edge_type_partition. -set (f:=(fun (e0 : SetRegReg.elt) (s0 : regregpartition) => - match env (fst e0) with - | Tint => - match env (snd e0) with - | Tint => - (SetRegReg.add e0 (rr1 s0), rr2 s0, - Regset.add (fst e0) (Regset.add (snd e0) (rr3 s0)), - rr4 s0) - | Tfloat => - (rr1 s0, rr2 s0, Regset.add (fst e0) (rr3 s0), - Regset.add (snd e0) (rr4 s0)) - end - | Tfloat => - match env (snd e0) with - | Tint => - (rr1 s0, rr2 s0, Regset.add (snd e0) (rr3 s0), - Regset.add (fst e0) (rr4 s0)) - | Tfloat => - (rr1 s0, SetRegReg.add e0 (rr2 s0), rr3 s0, - Regset.add (fst e0) (Regset.add (snd e0) (rr4 s0))) - end - end)). -unfold regregpartition in *. fold f. - -generalize (SetRegReg.empty,SetRegReg.empty, Regset.empty, Regset.empty). -generalize SetRegReg.elements_1. intro HH. -generalize (HH s e H). clear H HH. intro HH. -intro p. rewrite SetRegReg.fold_1. generalize p. clear p. -induction (SetRegReg.elements s). -inversion HH. -inversion HH. - subst. intro p. do 3 destruct p. simpl. - -assert (f a (t2,t3,t1,t0) = (t2, SetRegReg.add a t3, t1, Regset.add (fst a) (Regset.add (snd a) t0))). -unfold f. - -destruct H2. rewrite H in *. rewrite H2 in *. rewrite H0. rewrite H1. simpl. reflexivity. -rewrite H. -destruct H2. - -assert (forall x s1 s2 s3 s4, SetRegReg.In x s2 -> - SetRegReg.In x (rr2 (fold_left - (fun (a0 : SetRegReg.t*SetRegReg.t*Regset.t*Regset.t) (e0 : SetRegReg.elt) => f e0 a0) - l (s1, s2, s3, s4)))). - -clear H H0 H1 H2 HH IHl. - -induction l. -simpl. auto. -intros x s1 s2 s3 s4 H2. -simpl. - -assert (f a0 (s1,s2,s3,s4) = (SetRegReg.add a0 s1, s2,Regset.add (fst a0) (Regset.add (snd a0) s3), s4) \/ - f a0 (s1,s2,s3,s4) = (s1, SetRegReg.add a0 s2, s3, Regset.add (fst a0)(Regset.add (snd a0) s4)) \/ - f a0 (s1,s2,s3,s4) = (s1,s2,Regset.add (fst a0) s3, Regset.add (snd a0) s4)\/ - f a0 (s1,s2,s3,s4) = (s1,s2,Regset.add (snd a0) s3, Regset.add (fst a0) s4)). - -unfold f. -destruct (env (fst a0)); destruct (env (snd a0)); -unfold rr1,rr2,rr3,rr4; simpl; auto. - -destruct H. - -rewrite H. apply IHl. assumption. - -destruct H. -rewrite H. -apply IHl. apply SetRegReg.add_2. assumption. -destruct H; rewrite H. -apply IHl. assumption. -apply IHl. assumption. - -apply H4. apply SetRegReg.add_1. intuition. - -subst. simpl. intro p. apply IHl. assumption. -Qed. - -Lemma interf_int_regreg_translation : - interf_reg_reg (snd (fst (fst (Typed_interfgraphs g env)))) = - rr1 (regreg_edge_type_partition (interf_reg_reg g) env). - -Proof. -unfold Typed_interfgraphs. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. reflexivity. -Qed. - -Lemma interf_float_regreg_translation : - interf_reg_reg (snd (Typed_interfgraphs g env)) = - rr2 (regreg_edge_type_partition (interf_reg_reg g) env). - -Proof. -unfold Typed_interfgraphs. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. reflexivity. -Qed. - -Lemma correct_alloc_1 : check_coloring_1 g (my_graph_coloring g env) = true. - -Proof. -unfold check_coloring_1. -apply SetRegReg.for_all_1. - -unfold compat_bool. -intros x y H. destruct H as [H H0]. -rewrite H. rewrite H0. reflexivity. - -unfold SetRegReg.For_all. -intros x H. -generalize (Loc_reg_eq_type (fst x)). generalize (Loc_reg_eq_type (snd x)). -unfold my_graph_coloring in *. -change (snd (fst (fst (Typed_interfgraphs g env)))) with intG. -change (fst (fst (fst (Typed_interfgraphs g env)))) with intR. -change (snd (fst (Typed_interfgraphs g env))) with floatR. -change (snd (Typed_interfgraphs g env)) with floatG. -fold int_graph; fold float_graph. -unfold graph_coloring_aux. -change (map_to_coloring (IRC_map (graph_to_IRC_graph int_graph int_palette))) with - (IRC int_graph int_palette). -change (map_to_coloring (IRC_map (graph_to_IRC_graph float_graph float_palette))) with - (IRC float_graph float_palette). -intros Locty1 Locty2. -case_eq (Regs.get_type (Regs.P (fst x)) env); intros HH. -case_eq (Regs.get_type (Regs.P (snd x)) env); intros HH0. -case_eq (IRC int_graph int_palette (Regs.P (fst x))); intros. -destruct (register_heuristic_mreg (Regs.P (fst x)) t0 H0). rewrite H1. simpl. -case_eq (IRC int_graph int_palette (Regs.P (snd x))); intros. -destruct (register_heuristic_mreg (Regs.P (snd x)) t1 H2). rewrite H3. - -generalize (proper_coloring_IRC_aux int_graph int_palette (correct_palette_int)). -intro H4. unfold proper_coloring in H4. -destruct H4 as [H4 _]. -unfold proper_coloring_1 in H4. -assert (~Regs.eq (Regs.M x0) (Regs.M x1)). -apply (H4 (Regs.P (fst x), Regs.P (snd x), None)). -unfold Edge.interf_edge. auto. -unfold int_graph. -right. simpl. -apply regreg_IE_translation. unfold intG, typed_graphs. -rewrite interf_int_regreg_translation. -apply regreg_in_fst_partition. destruct x. auto. auto. auto. -change_rewrite. rewrite H0. rewrite H1. apply OptionReg.eq_refl. -change_rewrite. rewrite H2. rewrite H3. apply OptionReg.eq_refl. -destruct (Loc.eq (R x0) (R x1)). subst. -elim H5. inversion e. auto. -reflexivity. -destruct (Loc.eq (R x0) (S (Local (Zpos (snd x)) Tint))). -inversion e. -reflexivity. -case_eq (IRC int_graph int_palette (Regs.P (snd x))); intros. -destruct (register_heuristic_mreg (Regs.P (snd x)) t0 H1). rewrite H2. -destruct (Loc.eq (S (Local (Zpos (fst x)) Tint)) (R x0)). -inversion e. -reflexivity. -destruct (Loc.eq (S (Local (Zpos (fst x)) Tint)) (S (Local (Zpos (snd x)) Tint))). -inversion e. -elim (set_reg_reg_diff_ext _ _ (or_introl _ H) H3). -reflexivity. - -rewrite HH in *. rewrite HH0 in *. -set (l1 := match IRC int_graph int_palette (Regs.P (fst x)) with - | Some (Regs.P _) => S (Local 0 Tint) - | Some (Regs.M z) => R z - | None => S (Local (Zpos (fst x)) Tint) - end) in *. -set (l2 := match IRC float_graph float_palette (Regs.P (snd x)) with - | Some (Regs.P _) => S (Local 0 Tfloat) - | Some (Regs.M z) => R z - | None => S (Local (Zpos (snd x)) Tfloat) - end) in *. -destruct (Loc.eq l1 l2). rewrite e in Locty2. rewrite <-Locty1 in Locty2. congruence. -reflexivity. - -case_eq (Regs.get_type (Regs.P (snd x)) env); intros HH0. - -rewrite HH in *. rewrite HH0 in *. -set (l1 :=match IRC float_graph float_palette (Regs.P (fst x)) with - | Some (Regs.P _) => S (Local 0 Tfloat) - | Some (Regs.M z) => R z - | None => S (Local (Zpos (fst x)) Tfloat) - end ) in *. -set (l2 := match IRC int_graph int_palette (Regs.P (snd x)) with - | Some (Regs.P _) => S (Local 0 Tint) - | Some (Regs.M z) => R z - | None => S (Local (Zpos (snd x)) Tint) - end) in *. -destruct (Loc.eq l1 l2). rewrite e in Locty2. rewrite <-Locty1 in Locty2. congruence. -reflexivity. - -case_eq (IRC float_graph float_palette (Regs.P (fst x))); intros. -destruct (register_heuristic_mreg_float (Regs.P (fst x)) t0 H0). rewrite H1. simpl. -case_eq (IRC float_graph float_palette (Regs.P (snd x))); intros. -destruct (register_heuristic_mreg_float (Regs.P (snd x)) t1 H2). rewrite H3. - -generalize (proper_coloring_IRC_aux float_graph float_palette (correct_palette_float)). -intro H4. unfold proper_coloring in H4. -destruct H4 as [H4 _]. -unfold proper_coloring_1 in H4. -assert (~Regs.eq (Regs.M x0) (Regs.M x1)). -apply (H4 (Regs.P (fst x), Regs.P (snd x), None)). -unfold Edge.interf_edge. auto. -unfold float_graph. -right. simpl. -apply regreg_IE_translation. unfold floatG, typed_graphs. -rewrite interf_float_regreg_translation. -apply regreg_in_snd_partition. destruct x. auto. auto. auto. -change_rewrite. rewrite H0. rewrite H1. apply OptionReg.eq_refl. -change_rewrite. rewrite H2. rewrite H3. apply OptionReg.eq_refl. -destruct (Loc.eq (R x0) (R x1)). subst. -elim H5. inversion e. auto. -reflexivity. -destruct (Loc.eq (R x0) (S (Local (Zpos (snd x)) Tfloat))). -inversion e. -reflexivity. -case_eq (IRC float_graph float_palette (Regs.P (snd x))); intros. -destruct (register_heuristic_mreg_float (Regs.P (snd x)) t0 H1). rewrite H2. -destruct (Loc.eq (S (Local (Zpos (fst x)) Tfloat)) (R x0)). -inversion e. -reflexivity. -destruct (Loc.eq (S (Local (Zpos (fst x)) Tfloat)) (S (Local (Zpos (snd x)) Tfloat))). -inversion e. -elim (set_reg_reg_diff_ext _ _ (or_introl _ H) H3). -reflexivity. -Qed. - -Lemma interf_int_regmreg_translation : - interf_reg_mreg (snd (fst (fst (Typed_interfgraphs g env)))) = - rm1 (regmreg_edge_type_partition (interf_reg_mreg g) env). - -Proof. -unfold Typed_interfgraphs. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. reflexivity. -Qed. - -Lemma interf_float_regmreg_translation : - interf_reg_mreg (snd (Typed_interfgraphs g env)) = - rm2 (regmreg_edge_type_partition (interf_reg_mreg g) env). - -Proof. -unfold Typed_interfgraphs. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. reflexivity. -Qed. - -Lemma regmreg_in_fst_partition : forall e s, -SetRegMreg.In e s -> -env (fst e) = Tint -> -mreg_type (snd e) = Tint -> -SetRegMreg.In e (rm1 (regmreg_edge_type_partition s env)). - -Proof. -intros. -unfold regmreg_edge_type_partition. -set (f := (fun (e0 : SetRegMreg.elt) (s0 : regmregpartition) => - match env (fst e0) with - | Tint => - match mreg_type (snd e0) with - | Tint => - (SetRegMreg.add e0 (rm1 s0), rm2 s0, - Regset.add (fst e0) (rm3 s0), rm4 s0, - MRegset.add (snd e0) (rm5 s0), rm6 s0) - | Tfloat => - (rm1 s0, rm2 s0, Regset.add (fst e0) (rm3 s0), rm4 s0, - rm5 s0, MRegset.add (snd e0) (rm6 s0)) - end - | Tfloat => - match mreg_type (snd e0) with - | Tint => - (rm1 s0, rm2 s0, rm3 s0, Regset.add (fst e0) (rm4 s0), - MRegset.add (snd e0) (rm5 s0), rm6 s0) - | Tfloat => - (rm1 s0, SetRegMreg.add e0 (rm2 s0), rm3 s0, - Regset.add (fst e0) (rm4 s0), rm5 s0, - MRegset.add (snd e0) (rm6 s0)) - end - end)). -unfold regmregpartition in *. fold f. - -generalize (SetRegMreg.empty, SetRegMreg.empty, Regset.empty, Regset.empty, MRegset.empty, MRegset.empty). -generalize SetRegMreg.elements_1. intro HH. -generalize (HH s e H). clear H HH. intro HH. -intro p. rewrite SetRegMreg.fold_1. generalize p. clear p. -induction (SetRegMreg.elements s). -inversion HH. -inversion HH. - subst. intro p. do 5 destruct p. simpl. - -assert (f a (t4,t5,t3,t2,t1,t0) = (SetRegMreg.add a t4, t5, Regset.add (fst a) t3, t2, MRegset.add (snd a) t1, t0)). -unfold f. - -destruct H2. rewrite H in *. rewrite H2 in *. rewrite H0. rewrite H1. simpl. reflexivity. -rewrite H. -destruct H2. - -assert (forall x s1 s2 s3 s4 s5 s6, SetRegMreg.In x s1 -> - SetRegMreg.In x (rm1 (fold_left - (fun (a0 : SetRegMreg.t * SetRegMreg.t * Regset.t * Regset.t *MRegset.t * MRegset.t) - (e0 : SetRegMreg.elt) => f e0 a0) - l (s1, s2, s3, s4, s5, s6)))). - -clear H H0 H1 H2 HH IHl. - -induction l. -simpl. auto. -intros x s1 s2 s3 s4 s5 s6 H2. -simpl. - -assert (f a0 (s1,s2, s3, s4, s5, s6) = (SetRegMreg.add a0 s1, s2, Regset.add (fst a0) s3, s4, MRegset.add (snd a0) s5, s6) \/ - f a0 (s1,s2,s3,s4,s5,s6) = (s1, SetRegMreg.add a0 s2, s3, Regset.add (fst a0) s4, s5, MRegset.add (snd a0) s6) \/ - f a0 (s1,s2,s3,s4,s5,s6) = (s1,s2,Regset.add (fst a0) s3, s4,s5,MRegset.add (snd a0) s6) \/ - f a0 (s1,s2,s3,s4,s5,s6) = (s1,s2,s3, Regset.add (fst a0) s4, MRegset.add (snd a0) s5, s6)). - -unfold f. -destruct (env (fst a0)); destruct (mreg_type (snd a0)); auto. - -destruct H. - -rewrite H. apply IHl. apply SetRegMreg.add_2. assumption. - -destruct H. -rewrite H. -apply IHl. assumption. -destruct H; rewrite H; apply IHl; assumption. - -apply H4. apply SetRegMreg.add_1. intuition. - -subst. simpl. intro p. apply IHl. assumption. -Qed. - -Lemma regmreg_in_snd_partition : forall e s, -SetRegMreg.In e s -> -env (fst e) = Tfloat -> -mreg_type (snd e) = Tfloat -> -SetRegMreg.In e (rm2 (regmreg_edge_type_partition s env)). - -Proof. -intros. -unfold regmreg_edge_type_partition. -set (f := (fun (e0 : SetRegMreg.elt) (s0 : regmregpartition) => - match env (fst e0) with - | Tint => - match mreg_type (snd e0) with - | Tint => - (SetRegMreg.add e0 (rm1 s0), rm2 s0, - Regset.add (fst e0) (rm3 s0), rm4 s0, - MRegset.add (snd e0) (rm5 s0), rm6 s0) - | Tfloat => - (rm1 s0, rm2 s0, Regset.add (fst e0) (rm3 s0), rm4 s0, - rm5 s0, MRegset.add (snd e0) (rm6 s0)) - end - | Tfloat => - match mreg_type (snd e0) with - | Tint => - (rm1 s0, rm2 s0, rm3 s0, Regset.add (fst e0) (rm4 s0), - MRegset.add (snd e0) (rm5 s0), rm6 s0) - | Tfloat => - (rm1 s0, SetRegMreg.add e0 (rm2 s0), rm3 s0, - Regset.add (fst e0) (rm4 s0), rm5 s0, - MRegset.add (snd e0) (rm6 s0)) - end - end)). -unfold regmregpartition in *. fold f. - -generalize (SetRegMreg.empty, SetRegMreg.empty, Regset.empty, Regset.empty, MRegset.empty, MRegset.empty). -generalize SetRegMreg.elements_1. intro HH. -generalize (HH s e H). clear H HH. intro HH. -intro p. rewrite SetRegMreg.fold_1. generalize p. clear p. -induction (SetRegMreg.elements s). -inversion HH. -inversion HH. - subst. intro p. do 5 destruct p. simpl. - -assert (f a (t4,t5,t3,t2,t1,t0) = (t4, SetRegMreg.add a t5, t3, Regset.add (fst a) t2, t1, MRegset.add (snd a) t0)). -unfold f. - -destruct H2. rewrite H in *. rewrite H2 in *. rewrite H0. rewrite H1. simpl. reflexivity. -rewrite H. -destruct H2. - -assert (forall x s1 s2 s3 s4 s5 s6, SetRegMreg.In x s2 -> - SetRegMreg.In x (rm2 (fold_left - (fun (a0 : SetRegMreg.t * SetRegMreg.t * Regset.t * Regset.t *MRegset.t * MRegset.t) - (e0 : SetRegMreg.elt) => f e0 a0) - l (s1, s2, s3, s4, s5, s6)))). - -clear H H0 H1 H2 HH IHl. - -induction l. -simpl. auto. -intros x s1 s2 s3 s4 s5 s6 H2. -simpl. - -assert (f a0 (s1,s2, s3, s4, s5, s6) = (SetRegMreg.add a0 s1, s2, Regset.add (fst a0) s3, s4, MRegset.add (snd a0) s5, s6) \/ - f a0 (s1,s2,s3,s4,s5,s6) = (s1, SetRegMreg.add a0 s2, s3, Regset.add (fst a0) s4, s5, MRegset.add (snd a0) s6) \/ - f a0 (s1,s2,s3,s4,s5,s6) = (s1,s2,Regset.add (fst a0) s3, s4,s5,MRegset.add (snd a0) s6) \/ - f a0 (s1,s2,s3,s4,s5,s6) = (s1,s2,s3, Regset.add (fst a0) s4, MRegset.add (snd a0) s5, s6)). - -unfold f. -destruct (env (fst a0)); destruct (mreg_type (snd a0)); auto. - -destruct H. - -rewrite H. apply IHl. assumption. - -destruct H. -rewrite H. -apply IHl. apply SetRegMreg.add_2. assumption. -destruct H; rewrite H; apply IHl; assumption. - -apply H4. apply SetRegMreg.add_1. intuition. - -subst. simpl. intro p. apply IHl. assumption. -Qed. - -Section fold_assoc_map. - -Variable A : Type. - -Lemma fold_left_compat_map : forall (f : ColorMap.t Regs.t -> A -> ColorMap.t Regs.t) l e e', -ColorMap.Equal e e' -> -(forall e1 e2 a, ColorMap.Equal e1 e2 -> ColorMap.Equal (f e1 a) (f e2 a)) -> -ColorMap.Equal (fold_left f l e) (fold_left f l e'). - -Proof. -intro f;induction l;simpl. -auto. -intros e e' H H0 H1. -apply (IHl (f e a) (f e' a)). -apply H0;assumption. -assumption. -Qed. - -Lemma fold_left_assoc_map : forall l (f : ColorMap.t Regs.t -> A -> ColorMap.t Regs.t) x h, -(forall (y z : A) s, ColorMap.Equal (f (f s y) z) (f (f s z) y)) -> -(forall e1 e2 a, ColorMap.Equal e1 e2 -> ColorMap.Equal (f e1 a) (f e2 a)) -> -ColorMap.Equal (fold_left f (h :: l) x) (f (fold_left f l x) h). - -Proof. -induction l;simpl;intros f x h H H0. -intuition. -rewrite <-IHl;simpl;try assumption. -apply fold_left_compat_map;[apply H|];auto. -Qed. - -End fold_assoc_map. - -Lemma mreg_refl_coloring_aux : forall x gpalette, -VertexSet.In x (precolored (irc_g gpalette)) -> -VertexSet.Subset (precolored (irc_g gpalette)) (pal gpalette) -> -OptionReg.eq (map_to_coloring (IRC_map gpalette) x) (Some x). - -Proof. -intros. functional induction IRC_map gpalette; simpl in *. - -(* simplify *) -generalize (simplify_inv _ _ e). intro. -generalize (simplify_inv2 _ _ e). intro. destruct H2. simpl in *. clear e. -rewrite H2 in *. clear H2. unfold available_coloring. -set (palette := pal g0) in *. set (wl := irc_wl g0) in *. set (g1 := irc_g g0) in *. -case_eq ( VertexSet.choose - (VertexSet.diff palette - (forbidden_colors r - (IRC_map - (simplify_irc r g0 - (VertexSet.choose_1 (s:=get_simplifyWL wl) x0))) g1))). -intros. unfold map_to_coloring. -rewrite ColFacts.add_neq_o. -apply IHt0. unfold simplify_irc. simpl. -rewrite precolored_remove_vertex. apply VertexSet.remove_2. -intro. rewrite <-H3 in H. -generalize (In_simplify_props _ _ _ _ _ _ _ _ (VertexSet.choose_1 H1) (refl_equal _) (HWS_irc g0)). intro. -destruct H4. destruct H5. destruct H6. elim H7. auto. auto. -unfold simplify_irc. simpl. rewrite precolored_remove_vertex. -intro. intro. apply H0. apply (VertexSet.remove_3 H3). -intro. rewrite <-H3 in H. -generalize (In_simplify_props _ _ _ _ _ _ _ _ (VertexSet.choose_1 H1) (refl_equal _) (HWS_irc g0)). intro. -destruct H4. destruct H5. destruct H6. elim H7. auto. intro. -apply IHt0. unfold simplify_irc. simpl. -rewrite precolored_remove_vertex. apply VertexSet.remove_2. -intro. rewrite <-H3 in H. -generalize (In_simplify_props _ _ _ _ _ _ _ _ (VertexSet.choose_1 H1) (refl_equal _) (HWS_irc g0)). intro. -destruct H4. destruct H5. destruct H6. elim H7. auto. auto. -unfold simplify_irc. simpl. rewrite precolored_remove_vertex. -intro. intro. apply H0. apply (VertexSet.remove_3 H3). - -(* coalesce *) -assert (forall e', EdgeSet.In e' (get_movesWL (irc_wl g0)) -> In_graph_edge e' (irc_g g0)). -intros. -generalize (In_move_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (HWS_irc g0)). -intuition. -generalize (coalesce_inv _ _ e0). simpl. intro. -generalize (coalesce_inv_2 _ _ e0). intro. destruct H3. destruct H3. simpl in H3. rewrite H3 in *. clear H3. -generalize (any_coalescible_edge_1 _ _ _ _ H1 H2). -intro. destruct H3. -rewrite <-(moves_AE _ _ _ (HWS_irc g0)) in H4. -generalize (proj2 (proj1 (In_graph_aff_edge_in_AE _ _) H4)). intro. -generalize (any_coalescible_edge_2 _ _ _ _ H1 H2). intro. -unfold complete_coloring. -case_eq (ColorMap.find (elt:=Regs.t) (fst_ext e1) - (IRC_map (merge_irc e1 g0 x0 x1))). -intros. -unfold map_to_coloring. -rewrite ColFacts.add_neq_o. -apply IHt0. -assert (Edge.aff_edge e1). -rewrite (moves_AE _ _ _ (HWS_irc g0)) in H4. -generalize (In_move_props _ _ _ _ _ _ _ _ H4 (refl_equal _) (HWS_irc g0)). -intuition. -unfold merge_irc. simpl. -rewrite (precolored_merge _ _ H5 H8 _). -apply VertexSet.remove_2. intro. rewrite H9 in H6. elim H6. auto. auto. -unfold VertexSet.Subset in *. -intros. unfold merge_irc in *. simpl in *. -apply H0. rewrite precolored_merge in H8. apply (VertexSet.remove_3 H8). -intro. rewrite H8 in H6. elim H6. auto. -intro. -apply IHt0. -assert (Edge.aff_edge e1). -rewrite (moves_AE _ _ _ (HWS_irc g0)) in H4. -generalize (In_move_props _ _ _ _ _ _ _ _ H4 (refl_equal _) (HWS_irc g0)). -intuition. -unfold merge_irc. simpl. -rewrite (precolored_merge _ _ H5 H8 _). -apply VertexSet.remove_2. intro. rewrite H9 in H6. elim H6. auto. auto. -unfold VertexSet.Subset in *. -intros. unfold merge_irc in *. simpl in *. -apply H0. rewrite precolored_merge in H8. apply (VertexSet.remove_3 H8). - -(* freeze *) -generalize (freeze_inv _ _ e1). intro. -generalize (freeze_inv2 _ _ e1). intro. destruct H2. destruct H2. simpl in *. clear e1. -rewrite H2 in *. clear H2. unfold delete_preference_edges_irc2 in *. simpl in *. -apply IHt0. -rewrite precolored_delete_preference_edges. assumption. -unfold VertexSet.Subset in *. -intros. rewrite precolored_delete_preference_edges in H2. auto. - -(* spill *) -generalize e2. clear e e0 e1 e2. intro e. -generalize (spill_inv _ _ e). intro. -generalize (spill_inv2 _ _ e). intro. destruct H2. simpl in *. clear e. -rewrite H2 in *. clear H2. unfold available_coloring. -set (palette := pal g0) in *. set (wl := irc_wl g0) in *. set (g1 := irc_g g0) in *. -case_eq ( VertexSet.choose - (VertexSet.diff palette - (forbidden_colors r - (IRC_map - (spill_irc r g0 - (lowest_cost_in r (get_spillWL wl) g1 x0))) g1))). -intros. unfold map_to_coloring. -rewrite ColFacts.add_neq_o. -apply IHt0. unfold spill_irc. simpl. -rewrite precolored_remove_vertex. apply VertexSet.remove_2. -intro. rewrite <-H3 in H. -generalize (In_spill_props _ _ _ _ _ _ _ _ (lowest_cost_in _ _ _ H1) (refl_equal _) (HWS_irc g0)). intro. -destruct H4. destruct H5. elim H6. auto. auto. -unfold spill_irc. simpl. rewrite precolored_remove_vertex. -intro. intro. apply H0. apply (VertexSet.remove_3 H3). -intro. rewrite <-H3 in H. -generalize (In_spill_props _ _ _ _ _ _ _ _ (lowest_cost_in _ _ _ H1) (refl_equal _) (HWS_irc g0)). intro. -destruct H4. destruct H5. elim H6. auto. intro. -apply IHt0. unfold spill_irc. simpl. -rewrite precolored_remove_vertex. apply VertexSet.remove_2. -intro. rewrite <-H3 in H. -generalize (In_spill_props _ _ _ _ _ _ _ _ (lowest_cost_in _ _ _ H1) (refl_equal _) (HWS_irc g0)). intro. -destruct H4. destruct H5. elim H6. auto. auto. -unfold simplify_irc. simpl. rewrite precolored_remove_vertex. -intro. intro. apply H0. apply (VertexSet.remove_3 H3). - -(* ending case *) -set (palette := pal g0) in *. -set (g1 := irc_g g0) in *. -assert (map_to_coloring (precoloring_map g1) x = Some x). -unfold precoloring_map. -rewrite VertexSet.fold_1. -generalize VertexSet.elements_1. intro HH. -generalize (HH (precolored g1) x H). clear HH. intro HH. -generalize (NoDupA_elements (precolored g1)). intro HHH. -induction (VertexSet.elements (precolored g1)). -simpl. inversion HH. - -unfold map_to_coloring. -rewrite fold_left_assoc_map. -inversion HH. subst. -rewrite ColFacts.add_eq_o. -inversion H2; subst; auto. -apply Regs.eq_sym. auto. -subst. -unfold map_to_coloring in IHl. unfold Regs.t. -rewrite ColFacts.add_neq_o. -apply IHl. assumption. inversion HHH. assumption. -inversion HHH. subst. intro H6. -elim H4. -inversion H6; subst; auto. - -intros. -unfold ColorMap.Equal. -intro. -destruct (Regs.eq_dec z y0). -rewrite ColFacts.add_eq_o. -destruct (Regs.eq_dec y y0). -rewrite ColFacts.add_eq_o. -inversion e3; inversion e4; subst; rewrite H6; auto. -auto. -rewrite ColFacts.add_neq_o. -rewrite ColFacts.add_eq_o. -reflexivity. -assumption. -assumption. -assumption. -rewrite ColFacts.add_neq_o. -destruct (Regs.eq_dec y y0). -rewrite ColFacts.add_eq_o. -rewrite ColFacts.add_eq_o. -reflexivity. -assumption. -assumption. -rewrite ColFacts.add_neq_o. -rewrite ColFacts.add_neq_o. -rewrite ColFacts.add_neq_o. -reflexivity. -assumption. -assumption. -assumption. -assumption. - -intros. -apply ColFacts.add_m. -apply Regs.eq_refl. -reflexivity. -assumption. - -rewrite H1. apply OptionReg.eq_refl. -Qed. - -Lemma mreg_refl_coloring : forall x g palette, -VertexSet.In x (precolored g) -> -VertexSet.Subset (precolored g) palette -> -OptionReg.eq (IRC g palette x) (Some x). - -Proof. -intros. -apply mreg_refl_coloring_aux; -unfold graph_to_IRC_graph; simpl; auto. -Qed. - -Lemma loc_type_reg_type_equiv : forall x, -Loc.type (R x) = Regs.get_type (Regs.M x) env. - -Proof. -intro x. -unfold Loc.type. unfold Regs.get_type. reflexivity. -Qed. - -Lemma correct_alloc_2 : check_coloring_2 g (my_graph_coloring g env) = true. - -Proof. -unfold check_coloring_2. -apply SetRegMreg.for_all_1. - -unfold compat_bool. -intros x y H. destruct H as [H H0]. -rewrite H. rewrite H0. reflexivity. - -unfold SetRegMreg.For_all. -intros x H. -generalize (Loc_reg_eq_type (fst x)). generalize (loc_type_reg_type_equiv (snd x)). -unfold my_graph_coloring in *. -change (snd (fst (fst (Typed_interfgraphs g env)))) with intG. -change (fst (fst (fst (Typed_interfgraphs g env)))) with intR. -change (snd (fst (Typed_interfgraphs g env))) with floatR. -change (snd (Typed_interfgraphs g env)) with floatG. -fold int_graph; fold float_graph. -unfold graph_coloring_aux. -change (map_to_coloring (IRC_map (graph_to_IRC_graph int_graph int_palette))) with - (IRC int_graph int_palette). -change (map_to_coloring (IRC_map (graph_to_IRC_graph float_graph float_palette))) with - (IRC float_graph float_palette). -intros Locty1 Locty2. -case_eq (Regs.get_type (Regs.P (fst x)) env); intros HH. rewrite HH in *. -case_eq (Regs.get_type (Regs.M (snd x)) env); intros HH0. -case_eq (IRC int_graph int_palette (Regs.P (fst x))); intros. rewrite H0 in *. -destruct (register_heuristic_mreg (Regs.P (fst x)) t0 H0). rewrite H1 in *. simpl. -case_eq (IRC int_graph int_palette (Regs.M (snd x))); intros. -destruct (register_heuristic_mreg (Regs.M (snd x)) t1 H2). - -generalize (proper_coloring_IRC_aux int_graph int_palette (correct_palette_int)). -intro H4. unfold proper_coloring in H4. -destruct H4 as [H4 _]. -unfold proper_coloring_1 in H4. -assert (~Regs.eq (Regs.M x0) (Regs.M x1)). -apply (H4 (Regs.P (fst x), Regs.M (snd x), None)). -unfold Edge.interf_edge. auto. -unfold int_graph. -right. simpl. -apply regmreg_IE_translation. unfold intG, typed_graphs. -rewrite interf_int_regmreg_translation. -apply regmreg_in_fst_partition. destruct x. auto. auto. auto. -change_rewrite. rewrite H0. apply OptionReg.eq_refl. -change_rewrite. rewrite H2. rewrite H3. apply OptionReg.eq_refl. -destruct (Loc.eq (R x0) (R x1)). subst. -elim H5. inversion e. auto. - -destruct (Loc.eq (R x0) (R (snd x))). inversion e. clear e. -generalize (proper_coloring_IRC_aux int_graph int_palette correct_palette_int). -intro H6. unfold proper_coloring in H6. -destruct H6 as [H6 HH5]. destruct HH5 as [HH5 _]. -unfold proper_coloring_1 in H6. -assert (~Regs.req t0 (Regs.M (snd x))). -apply (H6 (Regs.P (fst x), Regs.M (snd x), None)). -unfold Edge.interf_edge. auto. -unfold int_graph. unfold intG, typed_graphs. -right. simpl. -apply regmreg_IE_translation. simpl. -apply regmreg_in_fst_partition. destruct x. auto. -auto. -auto. -change_rewrite. rewrite H0. rewrite H1. apply OptionReg.eq_refl. -change_rewrite. apply mreg_refl_coloring. subst. -apply (proj2 (precolored_equiv _ _)). -unfold is_precolored. simpl. -split. auto. - -assert (EdgeSet.In (Regs.reg_to_Reg (fst x), Regs.mreg_to_Reg (snd x), None) - (IE int_graph)). -unfold int_graph. -apply regmreg_IE_translation. destruct x. simpl. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. -replace p2 with (fst (regmreg_edge_type_partition (interf_reg_mreg g) env)). -apply regmreg_in_fst_partition. auto. -assumption. -assumption. -rewrite H7. auto. -apply (proj2 (In_graph_edge_in_ext (Regs.P (fst x), Regs.M (snd x), None) - _ (or_intror _ H1))). -assumption. - -subst. elim H8. apply Regs.eq_refl. reflexivity. - -assert (OptionReg.eq (IRC int_graph int_palette (Regs.M (snd x))) (Some (Regs.M (snd x)))) as Hsnd. -apply mreg_refl_coloring. subst. -apply (proj2 (precolored_equiv _ _)). -unfold is_precolored. simpl. -split. auto. - -assert (EdgeSet.In (Regs.reg_to_Reg (fst x), Regs.mreg_to_Reg (snd x), None) - (IE int_graph)). -unfold int_graph. -apply regmreg_IE_translation. destruct x. simpl. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. -replace p2 with (fst (regmreg_edge_type_partition (interf_reg_mreg g) env)). -apply regmreg_in_fst_partition. auto. -assumption. -assumption. -rewrite H4. auto. -apply (proj2 (In_graph_edge_in_ext (Regs.P (fst x), Regs.M (snd x), None) - _ (or_intror _ H1))). -assumption. - -rewrite H2 in Hsnd. inversion Hsnd. - -destruct (Loc.eq (S (Local (Zpos (fst x)) Tint)) (R (snd x))). -inversion e. reflexivity. - -case_eq (IRC int_graph int_palette (Regs.P (fst x))); intros. rewrite H0 in Locty2. -case_eq t0; intros; rewrite H1 in *. -destruct (Loc.eq (S (Local 0 Tint)) (R (snd x))). -inversion e. reflexivity. -destruct (Loc.eq (R m) (R (snd x))). -inversion e. subst. unfold Regs.get_type in HH0. unfold Loc.type in Locty2. congruence. -reflexivity. -destruct (Loc.eq (S (Local (Zpos (fst x)) Tint)) (R (snd x))). -inversion e. reflexivity. - -case_eq (Regs.get_type (Regs.M (snd x)) env); intros HH0. - -case_eq (IRC float_graph float_palette (Regs.P (fst x))); intros. rewrite H0 in Locty2. -case_eq t0; intros; rewrite H1 in *. -destruct (Loc.eq (S (Local 0 Tfloat)) (R (snd x))). -inversion e. reflexivity. -destruct (Loc.eq (R m) (R (snd x))). -inversion e. subst. rewrite HH in Locty2. unfold Regs.get_type in HH0. unfold Loc.type in Locty2. congruence. -reflexivity. -destruct (Loc.eq (S (Local (Zpos (fst x)) Tfloat)) (R (snd x))). -inversion e. reflexivity. - -case_eq (IRC float_graph float_palette (Regs.P (fst x))); intros. rewrite H0 in *. -destruct (register_heuristic_mreg_float (Regs.P (fst x)) t0 H0). rewrite H1 in *. simpl. -case_eq (IRC float_graph float_palette (Regs.M (snd x))); intros. -destruct (register_heuristic_mreg_float (Regs.M (snd x)) t1 H2). - -generalize (proper_coloring_IRC_aux float_graph float_palette (correct_palette_float)). -intro H4. unfold proper_coloring in H4. -destruct H4 as [H4 _]. -unfold proper_coloring_1 in H4. -assert (~Regs.eq (Regs.M x0) (Regs.M x1)). -apply (H4 (Regs.P (fst x), Regs.M (snd x), None)). -unfold Edge.interf_edge. auto. -unfold float_graph. -right. simpl. -apply regmreg_IE_translation. unfold floatG, typed_graphs. -rewrite interf_float_regmreg_translation. -apply regmreg_in_snd_partition. destruct x. auto. auto. auto. -change_rewrite. rewrite H0. apply OptionReg.eq_refl. -change_rewrite. rewrite H2. rewrite H3. apply OptionReg.eq_refl. -destruct (Loc.eq (R x0) (R x1)). subst. -elim H5. inversion e. auto. - -destruct (Loc.eq (R x0) (R (snd x))). inversion e. clear e. -generalize (proper_coloring_IRC_aux float_graph float_palette correct_palette_float). -intro H6. unfold proper_coloring in H6. -destruct H6 as [H6 HH5]. destruct HH5 as [HH5 _]. -unfold proper_coloring_1 in H6. -assert (~Regs.req t0 (Regs.M (snd x))). -apply (H6 (Regs.P (fst x), Regs.M (snd x), None)). -unfold Edge.interf_edge. auto. -unfold int_graph. unfold floatG, typed_graphs. -right. simpl. -apply regmreg_IE_translation. simpl. -apply regmreg_in_snd_partition. destruct x. auto. -auto. -auto. -change_rewrite. rewrite H0. rewrite H1. apply OptionReg.eq_refl. -change_rewrite. apply mreg_refl_coloring. subst. -apply (proj2 (precolored_equiv _ _)). -unfold is_precolored. simpl. -split. auto. - -assert (EdgeSet.In (Regs.reg_to_Reg (fst x), Regs.mreg_to_Reg (snd x), None) - (IE float_graph)). -unfold int_graph. -apply regmreg_IE_translation. destruct x. simpl. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. -replace p2 with (fst (regmreg_edge_type_partition (interf_reg_mreg g) env)). -apply regmreg_in_snd_partition. auto. -assumption. -assumption. -rewrite H7. auto. -apply (proj2 (In_graph_edge_in_ext (Regs.P (fst x), Regs.M (snd x), None) - _ (or_intror _ H1))). -assumption. - -subst. elim H8. apply Regs.eq_refl. reflexivity. - -assert (OptionReg.eq (IRC float_graph float_palette (Regs.M (snd x))) (Some (Regs.M (snd x)))) as Hsnd. -apply mreg_refl_coloring. subst. -apply (proj2 (precolored_equiv _ _)). -unfold is_precolored. simpl. -split. auto. - -assert (EdgeSet.In (Regs.reg_to_Reg (fst x), Regs.mreg_to_Reg (snd x), None) - (IE float_graph)). -unfold int_graph. -apply regmreg_IE_translation. destruct x. simpl. -case_eq (regreg_edge_type_partition (interf_reg_reg g) env); intros. -case_eq (regreg_edge_type_partition (pref_reg_reg g) env); intros. -case_eq (regmreg_edge_type_partition (interf_reg_mreg g) env); intros. -case_eq (regmreg_edge_type_partition (pref_reg_mreg g) env); intros. -simpl. -replace p2 with (fst (regmreg_edge_type_partition (interf_reg_mreg g) env)). -apply regmreg_in_snd_partition. auto. -assumption. -assumption. -rewrite H4. auto. -apply (proj2 (In_graph_edge_in_ext (Regs.P (fst x), Regs.M (snd x), None) - _ (or_intror _ H1))). -assumption. - -rewrite H2 in Hsnd. inversion Hsnd. - -destruct (Loc.eq (S (Local (Zpos (fst x)) Tfloat)) (R (snd x))). -inversion e. reflexivity. -Qed. - -Import Registers. - -Lemma in_palette_not_in_temporaries : forall x, -VertexSet.In (Regs.M x) int_palette -> -~In (R x) temporaries. - -Proof. -unfold int_palette. intros x H. -repeat (destruct (proj1 (Props.Dec.F.add_iff _ _ _) H);[ - inversion H0; subst; intro H1; inversion H1; - [inversion H2| repeat (destruct H2 as [H2|H2];[inversion H2|])]; - assumption - | generalize H0; clear H H0; intro H]). -elim (VertexSet.empty_1 H). -Qed. - -Lemma in_palette_not_in_temporaries_float : forall x, -VertexSet.In (Regs.M x) float_palette -> -~In (R x) temporaries. - -Proof. -unfold int_palette. intros x H. -repeat (destruct (proj1 (Props.Dec.F.add_iff _ _ _) H);[ - inversion H0; subst; intro H1; inversion H1; - [inversion H2| repeat (destruct H2 as [H2|H2];[inversion H2|])]; - assumption - | generalize H0; clear H H0; intro H]). -elim (VertexSet.empty_1 H). -Qed. - -Lemma coloring_acceptable_loc : forall x, -loc_is_acceptable (my_graph_coloring g env x) = true. - -Proof. -intro x. -unfold loc_is_acceptable. -unfold my_graph_coloring in *. -change (snd (fst (fst (Typed_interfgraphs g env)))) with intG. -change (fst (fst (fst (Typed_interfgraphs g env)))) with intR. -change (snd (fst (Typed_interfgraphs g env))) with floatR. -change (snd (Typed_interfgraphs g env)) with floatG. -fold int_graph; fold float_graph. -unfold graph_coloring_aux. -change (map_to_coloring (IRC_map (graph_to_IRC_graph int_graph int_palette))) with - (IRC int_graph int_palette). -change (map_to_coloring (IRC_map (graph_to_IRC_graph float_graph float_palette))) with - (IRC float_graph float_palette). -case_eq (Regs.get_type (Regs.P x) env); intros. -case_eq (IRC int_graph int_palette (Regs.P x)); intros. -unfold IRC in *. -generalize (register_heuristic_mreg _ _ H0). intro H2. -destruct H2. rewrite H1. -destruct (List.In_dec Loc.eq (R x0) temporaries). -assert (~In (R x0) temporaries). -apply in_palette_not_in_temporaries. -rewrite <-H1. -generalize (proper_coloring_IRC_aux int_graph int_palette - (correct_palette_int)). intro H3. -unfold proper_coloring in H3. -do 2 destruct H3 as [_ H3]. -unfold proper_coloring_3 in H3. -apply (H3 (Regs.P x) t0). -unfold IRC in *. rewrite H0. apply OptionReg.eq_refl. -elim (H2 i). -reflexivity. -unfold IRC in *. auto. -case_eq (IRC float_graph float_palette (Regs.P x)); intros. -unfold IRC in *. -generalize (register_heuristic_mreg_float _ _ H0). intro H2. -destruct H2. rewrite H1. -destruct (List.In_dec Loc.eq (R x0) temporaries). -assert (~In (R x0) temporaries). -apply in_palette_not_in_temporaries_float. -rewrite <-H1. -generalize (proper_coloring_IRC_aux float_graph float_palette - correct_palette_float). intro H3. -unfold proper_coloring in H3. -do 2 destruct H3 as [_ H3]. -unfold proper_coloring_3 in H3. -apply (H3 (Regs.P x) t0). -unfold IRC in *. rewrite H0 in *. apply OptionReg.eq_refl. -elim (H2 i). -reflexivity. -unfold IRC in *. auto. -Qed. - -Lemma correct_alloc_3 : check_coloring_3 (all_interf_regs g) env (my_graph_coloring g env) = true. - -Proof. -unfold check_coloring_3. -apply Regset.for_all_1. - -unfold compat_bool. -intros. subst. reflexivity. - -unfold Regset.For_all. -intros x H. -rewrite coloring_acceptable_loc. simpl. -unfold same_typ. -rewrite <-Loc_reg_eq_type. -simpl. destruct (env x); reflexivity. - -Qed. - -Theorem correct_alloc : check_coloring g env (all_interf_regs g) (my_graph_coloring g env) = true. - -Proof. -unfold check_coloring. -rewrite correct_alloc_1. -rewrite correct_alloc_2. -rewrite correct_alloc_3. -auto. -Qed. - -End Coloring_to_allocation. - -Lemma precolored_sub_int_palette : forall x g env, -VertexSet.In x (precolored (int_graph g env)) -> VertexSet.In x int_palette. - -Proof. -Admitted. - -Lemma precolored_sub_float_palette : forall x g env, -VertexSet.In x (precolored (float_graph g env)) -> VertexSet.In x float_palette. - -Proof. -Admitted. - -Theorem allocation_correct : forall g env, -check_coloring g env (all_interf_regs g) (my_graph_coloring g env) = true. - -Proof. -intros. apply correct_alloc. -intros. apply (precolored_sub_int_palette x g env). assumption. -intros. apply (precolored_sub_float_palette x g env). assumption. -Qed. diff --git a/backend/MyRegisters.v b/backend/MyRegisters.v deleted file mode 100755 index c2c7460..0000000 --- a/backend/MyRegisters.v +++ /dev/null @@ -1,143 +0,0 @@ -Require Import OrderedType. -Require Import Registers. -Require Import NArith. -Require Import Locations. -Require Import InterfGraph. - -Definition op_plus (x y : option N) : option N := -match (x,y) with -|(None,None) => None -|(Some a,Some b) => Some (Nplus a b) -|(Some a,None) => Some a -|(None,Some b) => Some b -end. - -(* Definition of registers as Pseudo-registers or machine registers *) - -Module Import Regs <: OrderedType. - -Inductive registers : Type := -|P : reg -> registers -|M : mreg -> registers. - -Definition t := registers. - -Definition reg_to_Reg := fun r => P r. -Definition mreg_to_Reg := fun r => M r. - -Inductive req : t -> t -> Prop := -|Peq : forall x y, eq x y -> req (P x) (P y) -|Meq : forall x y, eq x y -> req (M x) (M y). - -Definition eq := req. - -Inductive rlt : t -> t -> Prop := -|Plt : forall x y, OrderedReg.lt x y -> rlt (P x) (P y) -|Mlt : forall x y, OrderedMreg.lt x y -> rlt (M x) (M y) -|PMlt : forall x y, rlt (M x) (P y). - -Definition lt := rlt. - -Lemma eq_refl : forall x, eq x x. - -Proof. -induction x;constructor;reflexivity. -Qed. - -Lemma eq_sym : forall x y, eq x y -> eq y x. - -Proof. -induction 1;rewrite H;apply eq_refl. -Qed. - -Lemma eq_trans : forall x y z, eq x y -> eq y z -> eq x z. - -Proof. -intros x y z H H0. -inversion H;inversion H0;subst. -rewrite H5;apply eq_refl. -inversion H5. -inversion H5. -rewrite H5;apply eq_refl. -Qed. - -Lemma lt_trans : forall x y z, lt x y -> lt y z -> lt x z. - -Proof. -intros x y z H H0. -inversion H;inversion H0;subst. -inversion H5;subst. -constructor. apply (OrderedReg.lt_trans _ _ _ H1 H4). -inversion H5. -inversion H4. -inversion H5. -inversion H5;subst. -constructor. apply (OrderedMreg.lt_trans _ _ _ H1 H4). -constructor. -constructor. -inversion H4. -constructor. -Qed. - -Lemma lt_not_eq : forall x y, lt x y -> ~eq x y. - -Proof. -induction 1;intro H0;inversion H0. -elim (OrderedReg.lt_not_eq _ _ H H3). -elim (OrderedMreg.lt_not_eq _ _ H H3). -Qed. - -Lemma compare : forall x y, Compare lt eq x y. - -Proof. -intros x y. -destruct x;destruct y. -destruct (OrderedReg.compare r r0). -apply LT. constructor. assumption. -apply EQ. constructor. assumption. -apply GT. constructor. assumption. -apply GT. constructor. -apply LT. constructor. -destruct (OrderedMreg.compare m m0). -apply LT. constructor. assumption. -apply EQ. constructor. assumption. -apply GT. constructor. assumption. -Qed. - -Lemma eq_dec : forall x y, {eq x y}+{~eq x y}. - -Proof. -intros x y. -destruct (compare x y). -right. apply lt_not_eq. assumption. -left. assumption. -right. (cut (~ eq y x)). -intros H H0. elim H. apply eq_sym. assumption. -apply lt_not_eq. assumption. -Qed. - -Definition get_type x env := -match x with -| P y => env y -| M y => mreg_type y -end. - -End Regs. - -Definition is_mreg x := -match x with -| P _ => false -| M _ => true -end. - -Lemma mreg_ext : forall x y, -Regs.eq x y -> is_mreg x = is_mreg y. - -Proof. -intros. -destruct x; destruct y; simpl in *. -reflexivity. -inversion H. -inversion H. -reflexivity. -Qed. diff --git a/backend/Order_arith.v b/backend/Order_arith.v deleted file mode 100755 index 298468c..0000000 --- a/backend/Order_arith.v +++ /dev/null @@ -1,349 +0,0 @@ -Require Import Arith. -Require Import ZArith. - -(* Definition of some facts other arith, for the termination order... - many uninteresting but necessary lemmas *) - -Fixpoint puissance_aux x y res {struct y} : nat := -match y with -| 0 => res -| S n => puissance_aux x n (x*res) -end. - -Definition puissance x y := puissance_aux x y 1. - -Lemma dec_base_aux : forall a b c d x, -0 < a -> -a + b +c +d < x -> -a*puissance x 3 + b*puissance x 2 + -c*x + d > 0. - -Proof. -intros a b c d x Ha H. -induction a. -inversion Ha. -unfold puissance;simpl;destruct x. -inversion H. -apply lt_le_trans with (m:=S x * (S x * (S x *1))). -rewrite mult_1_r. -case_eq (S x * (S x * S x)). -intro H0. -inversion H0. -intros n H0. -intuition. -intuition. -Qed. - -Lemma inf_diff : forall x y, -x - y > 0 -> x > y. - -Proof. -intuition. -Qed. - -Lemma plop : forall x y, -y > 0 -> -x <= x*y. - -Proof. -induction y;intros. -inversion H. -rewrite mult_succ_r. -intuition. -Qed. - -Lemma mult_zero : forall x y, -x*y = 0 -> x = 0 \/ y = 0. - -Proof. -intros. -induction x. -left;reflexivity. -induction y. -right;reflexivity. -inversion H. -Qed. - -Lemma mult_plop : forall x y z, -x > 0 -> -y < z -> -y*x < z*x. - -Proof. -intros x y z H H0. -induction x. -inversion H. -destruct (lt_eq_lt_dec x 0). -destruct s. -inversion l. -do 2 rewrite mult_succ_r. -rewrite e. -do 2 rewrite mult_0_r. -intuition. -generalize (IHx l). -do 2 rewrite mult_succ_r. -intuition. -Qed. - -Lemma mult_plop_eq : forall x y z, -x > 0 -> -y <= z -> -y*x <= z*x. - -Proof. -intros x y z H H0. -induction x. -inversion H. -destruct (lt_eq_lt_dec x 0). -destruct s. -inversion l. -do 2 rewrite mult_succ_r. -rewrite e. -do 2 rewrite mult_0_r. -intuition. -generalize (IHx l). -do 2 rewrite mult_succ_r. -intuition. -Qed. - -Lemma mult_plop_eq2 : forall x y z, -x > 0 -> -y <= z -> -x*y <= x*z. - -Proof. -intros x y z H H0. -induction x. -inversion H. -destruct (lt_eq_lt_dec x 0). -destruct s. -inversion l. -do 2 rewrite mult_succ_l. -rewrite e. -do 2 rewrite mult_0_l. -intuition. -generalize (IHx l). -do 2 rewrite mult_succ_l. -intuition. -Qed. - -Lemma add_inf : forall x y z, -x <= y -> z+x <= z+y. - -Proof. -intuition. -Qed. - -Lemma add_hd_eq : forall x y z, -x = y -> z+x = z+y. - -Proof. -auto. -Qed. - -Lemma add_hd_inf : forall x y z, -x < y -> z+x < z + y. - -Proof. -intuition. -Qed. - -Lemma dec_base : forall a b c d a' b' c' d' x, -a + b + c + d < x -> x > 0 -> -(a < a' -> -(a *puissance x 3 +b *puissance x 2 + c *x + d < - a' *puissance x 3 +b' *puissance x 2 + c' *x + d')). - -Proof. -intros a b c d a' b' c' d' x H H0 H1. -unfold puissance;simpl. -rewrite mult_1_r. -apply lt_le_trans with (m := (S a)*x*x*x). -rewrite mult_succ_l. -do 2 rewrite mult_plus_distr_r. -replace (a*(x*(x*x))+b*(x*x)+c*x+d) with (a*x*x*x+(b*x*x+c*x+d)). -apply add_hd_inf. -assert (c*x <= c*x*x). -apply plop. -assumption. -assert (d <= d*x*x). -apply le_trans with (m := d*x). -apply plop. -assumption. -apply plop. -assumption. -apply le_lt_trans with (b*x*x+c*x*x+d*x*x). -intuition. -replace (b*x*x+c*x*x+d*x*x) with ((b+c+d)*x*x). -apply mult_plop. -assumption. -apply mult_plop. -assumption. -intuition. -do 4 rewrite mult_plus_distr_r. -reflexivity. -do 2 rewrite plus_assoc. -replace (a*(x* (x*x))) with (a*x*x*x). -replace (b*(x*x)) with (b*x*x). -reflexivity. -intuition. -replace (x*(x*x)) with (x*x*x). -rewrite mult_assoc. -intuition. -intuition. -apply le_trans with (m := a'*(x*(x*x))). -do 2 rewrite mult_assoc. -intuition. -apply mult_plop_eq. -assumption. -apply mult_plop_eq. -assumption. -apply mult_plop_eq. -assumption. -assumption. -intuition. -Qed. - -Lemma dec_base2 : forall a b c a' b' c' x, -a + b + c < x -> x > 0 -> -(a < a' -> -(a *puissance x 2 +b *x+ c < - a' *puissance x 2 +b' *x + c')). - -Proof. -intros a b c a' b' c' x H H0 H1. -unfold puissance;simpl. -rewrite mult_1_r. -apply lt_le_trans with (m := S a*x*x). -rewrite mult_succ_l. -rewrite mult_plus_distr_r. -rewrite mult_assoc. -rewrite <-plus_assoc. -apply add_hd_inf. -assert (c <= c*x). -apply plop. -assumption. -apply le_lt_trans with (m := b*x+c*x). -intuition. -replace (b*x+c*x) with ((b+c)*x). -apply mult_plop. -assumption. -intuition. -intuition. -apply le_trans with (m := a'*x*x). -apply mult_plop_eq. -assumption. -apply mult_plop_eq. -assumption. -intuition. -rewrite mult_assoc. -intuition. -Qed. - -Lemma dec_base3 : forall a b a' b' x, -a + b < x -> x > 0 -> a < a' --> (a *x +b < a' *x + b'). - -Proof. -intros a b a' b' x H H0 H1. -apply lt_le_trans with (m := S a*x). -rewrite mult_succ_l. -intuition. -assert (S a <= a'). -intuition. -apply le_trans with (m:= a'*x). -apply mult_plop_eq;assumption. -intuition. -Qed. - -Lemma puiss_aux : forall x y n, -puissance_aux x y n = n*puissance x y. - -Proof. -unfold puissance;induction y;simpl. -intuition. -intro n. -rewrite mult_1_r. -rewrite (IHy x). -rewrite (IHy (x*n)). -rewrite mult_assoc. -rewrite (mult_comm x n). -reflexivity. -Qed. - -Lemma puiss_step : forall x n, -puissance x (S n) = x*puissance x n. - -Proof. -intros. -unfold puissance;simpl. -rewrite puiss_aux;simpl. -intuition. -Qed. - -Lemma pos_puiss : forall x y, -x > 0 -> puissance x y > 0. - -Proof. -induction y;intros. -unfold puissance;simpl. -intuition. -rewrite puiss_step. -generalize (IHy H);intro H0. -case_eq (x*puissance x y);intros. -generalize (mult_zero _ _ H1);intros. -destruct H2;intuition. -intuition. -Qed. - -Lemma incr_puis : forall x y n, -x <= y -> -puissance x n <= puissance y n. - -Proof. -induction n;intros;simpl. -intuition. -do 2 rewrite puiss_step. -destruct x. -intuition. -destruct y. -inversion H. -apply le_trans with (m := S x*puissance (S y) n). -apply mult_plop_eq2. -intuition. -apply (IHn H). -apply mult_plop_eq. -apply pos_puiss. -intuition. -assumption. -Qed. - -Lemma incr_puis2 : forall x n p, -n <= p -> -x > 0 -> -puissance x n <= puissance x p. - -Proof. -induction n;intros. -unfold puissance;simpl. -fold (puissance x p). -generalize (pos_puiss). -intuition. -unfold puissance;simpl. -rewrite mult_1_r. -do 2 rewrite puiss_aux. -rewrite mult_1_l. -generalize (IHn (p-1));intro. -assert (puissance x n <= puissance x (p-1)). -apply H1. -intuition. -assumption. -destruct p. -inversion H. -rewrite puiss_step. -apply mult_plop_eq2. -assumption. -apply IHn. -intuition. -assumption. -Qed.
\ No newline at end of file diff --git a/backend/OrderedOption.v b/backend/OrderedOption.v deleted file mode 100755 index b2cebe2..0000000 --- a/backend/OrderedOption.v +++ /dev/null @@ -1,96 +0,0 @@ -Require Import FSets. - -(* Definition of options types as ordered types, - in order to define weights *) - -Module OrderedOpt (O : OrderedType) <: OrderedType. - -Definition t := option O.t. - -Inductive eq_ : t -> t -> Prop := -|None_eq : eq_ None None -|Some_eq : forall x y, O.eq x y -> eq_ (Some x) (Some y). - -Definition eq := eq_. - -Inductive lt_ : t -> t -> Prop := -|None_lt : forall o : O.t, lt_ None (Some o) -|Some_lt : forall o o', O.lt o o' -> lt_ (Some o) (Some o'). - -Definition lt := lt_. - -Lemma eq_refl : forall x : t, eq x x. - -Proof. -unfold eq;intro x;destruct x;constructor;apply O.eq_refl. -Qed. - -Lemma eq_sym : forall x y : t, eq x y -> eq y x. - -Proof. -unfold eq;intros x y H;destruct x;destruct y. -constructor;inversion H;apply O.eq_sym;assumption. -inversion H. -inversion H. -assumption. -Qed. - -Lemma eq_trans : forall x y z, eq x y -> eq y z -> eq x z. - -Proof. -unfold eq;intros x y z H H0. -inversion H;inversion H0. -constructor. -rewrite <-H2 in H4;inversion H4. -rewrite <-H3 in H4;inversion H4. -rewrite <-H3 in H5;inversion H5;subst. -constructor;apply (O.eq_trans H1 H4). -Qed. - -Lemma lt_trans : forall x y z, lt x y -> lt y z -> lt x z. - -Proof. -unfold lt;intros x y z H H0. -inversion H;inversion H0;constructor. -rewrite <-H3 in H4;inversion H4. -rewrite <-H3 in H5;inversion H5. -subst. -apply (O.lt_trans H1 H4). -Qed. - -Lemma lt_not_eq : forall x y, lt x y -> ~eq x y. - -Proof. -intros x y H. -unfold eq;intro Heq. -inversion H;inversion Heq. -rewrite <-H3 in H1;inversion H1. -rewrite <-H0 in H3;inversion H3. -rewrite <-H1 in H3;inversion H3. -subst;inversion H4;inversion H5;subst. -elim (O.lt_not_eq H0 H3). -Qed. - -Lemma compare : forall x y : t, Compare lt eq x y. - -Proof. -intros x y. -destruct x;destruct y. -destruct (O.compare t0 t1); -[apply LT;unfold lt|apply EQ;unfold eq|apply GT;unfold lt];constructor;assumption. -apply GT;unfold lt;constructor. -apply LT;unfold lt;constructor. -apply EQ;unfold eq;constructor. -Qed. - -Lemma eq_dec : forall x y, {eq x y}+{~eq x y}. - -Proof. -intros x y. -destruct (compare x y). -right. apply lt_not_eq. assumption. -left. assumption. -right. intro H. generalize (eq_sym _ _ H). intro H0. elim (lt_not_eq _ _ l H0). -Qed. - -End OrderedOpt.
\ No newline at end of file diff --git a/backend/Regs.v b/backend/Regs.v deleted file mode 100755 index 33e3fea..0000000 --- a/backend/Regs.v +++ /dev/null @@ -1,122 +0,0 @@ -Require Import Registers. - - -Definition op_plus (x y : option N) : option N := -match (x,y) with -|(None,None) => None -|(Some a,Some b) => Some (Nplus a b) -|(Some a,None) => Some a -|(None,Some b) => Some b -end. - -Module Regs <: OrderedType. - -(* Definition of registers as Pseudo-registers or machine registers *) - -Inductive registers : Type := -|P : reg -> registers -|M : mreg -> registers. - -Definition t := registers. - -Definition reg_to_Reg := fun r => P r. -Definition mreg_to_Reg := fun r => M r. - -Inductive req : t -> t -> Prop := -|Peq : forall x y, eq x y -> req (P x) (P y) -|Meq : forall x y, eq x y -> req (M x) (M y). - -Definition eq := req. - -Inductive rlt : t -> t -> Prop := -|Plt : forall x y, OrderedReg.lt x y -> rlt (P x) (P y) -|Mlt : forall x y, OrderedMreg.lt x y -> rlt (M x) (M y) -|PMlt : forall x y, rlt (M x) (P y). - -Definition lt := rlt. - -Lemma eq_refl : forall x, eq x x. - -Proof. -induction x;constructor;reflexivity. -Qed. - -Lemma eq_sym : forall x y, eq x y -> eq y x. - -Proof. -induction 1;rewrite H;apply eq_refl. -Qed. - -Lemma eq_trans : forall x y z, eq x y -> eq y z -> eq x z. - -Proof. -intros x y z H H0. -inversion H;inversion H0;subst. -rewrite H5;apply eq_refl. -inversion H5. -inversion H5. -rewrite H5;apply eq_refl. -Qed. - -Lemma lt_trans : forall x y z, lt x y -> lt y z -> lt x z. - -Proof. -intros x y z H H0. -inversion H;inversion H0;subst. -inversion H5;subst. -constructor. apply (OrderedReg.lt_trans H1 H4). -inversion H5. -inversion H4. -inversion H5. -inversion H5;subst. -constructor. apply (OrderedMreg.lt_trans _ _ _ H1 H4). -constructor. -constructor. -inversion H4. -constructor. -Qed. - -Lemma lt_not_eq : forall x y, lt x y -> ~eq x y. - -Proof. -induction 1;intro H0;inversion H0. -elim (OrderedReg.lt_not_eq H H3). -elim (OrderedMreg.lt_not_eq _ _ H H3). -Qed. - -Lemma compare : forall x y, Compare lt eq x y. - -Proof. -intros x y. -destruct x;destruct y. -destruct (OrderedReg.compare r r0). -apply LT. constructor. assumption. -apply EQ. constructor. assumption. -apply GT. constructor. assumption. -apply GT. constructor. -apply LT. constructor. -destruct (OrderedMreg.compare m m0). -apply LT. constructor. assumption. -apply EQ. constructor. assumption. -apply GT. constructor. assumption. -Qed. - -Lemma eq_dec : forall x y, {eq x y}+{~eq x y}. - -Proof. -intros x y. -destruct (compare x y). -right. apply lt_not_eq. assumption. -left. assumption. -right. (cut (~ eq y x)). -intros H H0. elim H. apply eq_sym. assumption. -apply lt_not_eq. assumption. -Qed. - -Definition get_type x env := -match x with -| P y => env y -| M y => mreg_type y -end. - -End Regs.
\ No newline at end of file diff --git a/backend/Remove_Vertex_Adjacency.v b/backend/Remove_Vertex_Adjacency.v deleted file mode 100755 index 668063d..0000000 --- a/backend/Remove_Vertex_Adjacency.v +++ /dev/null @@ -1,73 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Interference_adjacency. -Require Import Edges. -Require Import MyRegisters. - -Import RegFacts Props. - -Module Register := Regs. - -(* For any vertex x different from r, the neighborhood of x in - (remove_vertex r g) is its one in g, minus r *) -Lemma remove_interf_adj : forall x r g, -~Register.eq x r -> -VertexSet.Equal (interference_adj x (remove_vertex r g)) - (VertexSet.remove r (interference_adj x g)). - -Proof. -intros. -split; intros. -apply VertexSet.remove_2. intro. -rewrite <-H1 in H0. rewrite in_interf in H0. -generalize (proj1 (In_graph_edge_in_ext _ _ H0)). change_rewrite. intro. -rewrite In_remove_vertex in H2. destruct H2. elim (H3 (Register.eq_refl _)). -rewrite in_interf in H0. rewrite In_remove_edge in H0. destruct H0. -rewrite in_interf. assumption. - -rewrite in_interf. rewrite In_remove_edge. -split. -rewrite <-in_interf. apply (VertexSet.remove_3 H0). -intro. destruct H1; change_rewrite. -elim (VertexSet.remove_1 H1 H0). -elim H. auto. -Qed. - -(* If x is different from r and does not belong to the interference neighborhood - of r in g, then the interference neighborhood of x in (remove_vertex r g) - is equal to the interference neighborhood of x in g *) -Lemma interf_adj_remove : forall x r g, -~VertexSet.In x (interference_adj r g) -> -~Register.eq x r -> -VertexSet.Equal (interference_adj x g) (interference_adj x (remove_vertex r g)). - -Proof. -intros x r g H H0. -rewrite remove_interf_adj. rewrite remove_equal. apply VertexSet.eq_refl. -intro. elim H. apply interf_adj_comm. assumption. -auto. -Qed. - -(* The interference neighborhood of x in (remove_vertex r g) - is a subset of the interference neighborhood of x in g *) -Lemma sub_remove_interf : forall x r g, -~Register.eq x r -> -VertexSet.Subset (interference_adj x (remove_vertex r g)) - (interference_adj x g). - -Proof. -intros x r g H. rewrite remove_interf_adj. -unfold VertexSet.Subset. intros y H0. -apply (VertexSet.remove_3 H0). -assumption. -Qed. - -(* If x is a neighbor of r in g, then x belongs to (remove_vertex r g) *) -Lemma in_interf_adj_in_remove : forall x r g, -VertexSet.In x (interference_adj r g) -> In_graph x (remove_vertex r g). - -Proof. -intros x r g H. rewrite In_remove_vertex. split. -rewrite in_interf in H. apply (proj1 (In_graph_edge_in_ext _ _ H)). -intro H0. rewrite H0 in H. elim (not_in_interf_self _ _ H). -Qed. diff --git a/backend/Remove_Vertex_Degree.v b/backend/Remove_Vertex_Degree.v deleted file mode 100755 index e296265..0000000 --- a/backend/Remove_Vertex_Degree.v +++ /dev/null @@ -1,173 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Remove_Vertex_Adjacency. -Require Import ZArith. -Require Import Edges. -Require Import MyRegisters. -Require Import Interference_adjacency. - -Module Register := Regs. - -Import RegFacts Props. - -(* The degree of any vertex different from r decreases when - r is removed from the a graph g. Hence, a low-degree vertex of g - is a low-degree vertex of (remove_vertex r g) *) -Lemma low_remove_low : forall x r g K, -has_low_degree g K x = true -> -~Register.eq x r -> -has_low_degree (remove_vertex r g) K x = true. - -Proof. -intros x r g K H H0. unfold has_low_degree, interf_degree in *. -generalize (sub_remove_interf _ _ g H0);intro H1. -generalize (subset_cardinal H1);intro H2. -destruct (le_lt_dec K (VertexSet.cardinal (interference_adj x g))). -inversion H. -destruct (le_lt_dec K (VertexSet.cardinal (interference_adj x (remove_vertex r g)))). -apply False_ind;intuition. -auto. -Qed. - -(* For the same reason, a high-degree vertex of (remove_vertex r g) - is a high-degree vertex of g *) -Lemma not_low_remove_not_low : forall x r g K, -has_low_degree (remove_vertex r g) K x = false -> -~Register.eq x r -> -has_low_degree g K x = false. - -Proof. -intros x r g K H H0. -case_eq (has_low_degree g K x);[intros|auto]. -rewrite (low_remove_low _ _ _ _ H1) in H. inversion H. -auto. -Qed. - -(* The degree of any vertex x which is not an interference neighbor of - (remove_vertex r g) is the same in g and in (remove_vertex r g) *) -Lemma low_deg : forall x r g K, -~VertexSet.In x (interference_adj r g) -> -~Register.eq x r -> -has_low_degree g K x = has_low_degree (remove_vertex r g) K x. - -Proof. -intros x r g K H H0. unfold has_low_degree, interf_degree. -destruct (le_lt_dec K (VertexSet.cardinal (interference_adj x g))); -destruct (le_lt_dec K (VertexSet.cardinal (interference_adj x (remove_vertex r g)))). -reflexivity. -rewrite <-(cardinal_m(interf_adj_remove _ _ _ H H0)) in l0. -apply False_ind. intuition. -elim (lt_irrefl (VertexSet.cardinal (interference_adj x g))). -apply lt_le_trans with (m := K). auto. -rewrite (cardinal_m (interf_adj_remove x r g H H0)). auto. -reflexivity. -Qed. - -Lemma lt_le_S_eq : forall n x, -x < n -> -n <= S x -> -n = S x. - -Proof. -induction n; intros. -intuition. -induction x; intros. -apply eq_S. intuition. -apply eq_S. apply IHn. intuition. intuition. -Qed. - -Lemma le_S_irrefl : forall x, -S x <= x -> False. - -Proof. -induction x; intros. -inversion H. -apply IHx. apply le_S_n. assumption. -Qed. - -(* If x is a low-degree degree of (remove_vertex r g) and x is - a high-degree vertex of g then the interference degree - of x in g is exactly k *) -Lemma degree_dec_remove_K : forall x k r g, -~Register.eq x r -> -has_low_degree g k x = false -> -has_low_degree (remove_vertex r g) k x = true -> -interf_degree g x = k. - -Proof. -unfold has_low_degree, interf_degree. intros. -destruct (le_lt_dec k (VertexSet.cardinal (interference_adj x (remove_vertex r g)))) in H1. -inversion H1. -rewrite (cardinal_m (remove_interf_adj _ _ _ H)) in l. -unfold has_low_degree in H0. -destruct (le_lt_dec k (VertexSet.cardinal (interference_adj x g))). -destruct (In_dec r (interference_adj x g)). -generalize (remove_cardinal_1 i). intro HH. rewrite <-HH in l0. -set (card := VertexSet.cardinal (VertexSet.remove r (interference_adj x g))) in *. -rewrite <-HH. symmetry. apply lt_le_S_eq; auto. -generalize (remove_cardinal_2 n). intro HH. rewrite <-HH in l0. -elim (lt_irrefl k). intuition. -inversion H0. -Qed. - -(* Any x which is of high-degree in g and of low-degree in (remove_vertex r g) - interferes with r *) -Lemma low_degree_in_interf : forall x r g K, -has_low_degree (remove_vertex r g) K x = true -> -~Register.eq x r -> -has_low_degree g K x = false -> -VertexSet.In x (interference_adj r g). - -Proof. -unfold has_low_degree, interf_degree. intros x r g K H HH H0. -destruct (le_lt_dec K (VertexSet.cardinal (interference_adj x g))). -destruct (le_lt_dec K (VertexSet.cardinal (interference_adj x (remove_vertex r g)))). -inversion H. -destruct (In_dec x (interference_adj r g)). -assumption. -generalize (interf_adj_remove _ _ _ n HH);intro H2. -rewrite (cardinal_m H2) in l. -apply False_ind. intuition. -inversion H0. -Qed. - -(* Reciprocally, a high-degree vertex x of g which is - exactly of degree k in g and interferes with r is - of low-degree in (remove_vertex r g) *) - -Lemma degree_K_remove_dec : forall x k r g, -~Register.eq x r -> -interf_degree g x = k -> -VertexSet.In x (interference_adj r g) -> -has_low_degree (remove_vertex r g) k x = true. - -Proof. -unfold has_low_degree, interf_degree. intros. -destruct (le_lt_dec k (VertexSet.cardinal (interference_adj x g))). -destruct (le_lt_dec k (VertexSet.cardinal (interference_adj x (remove_vertex r g)))). -rewrite (cardinal_m (remove_interf_adj _ _ _ H)) in l0. -generalize (remove_cardinal_1 (interf_adj_comm _ _ _ H1)). -intro HH. rewrite <-HH in l. -set (card := VertexSet.cardinal (VertexSet.remove r (interference_adj x g))) in *. -rewrite <-HH in H0. rewrite <-H0 in *. elim (le_S_irrefl _ l0). -reflexivity. -apply False_ind. intuition. -Qed. - -(* Finally, an unused but meaningful theorem summarizing - conditions leading to an evolution of the interference degrees - when a vertex r is removed *) -Theorem remove_low_degree_evolution : forall x k r g, -~Register.eq x r -> -((has_low_degree g k x = false /\ has_low_degree (remove_vertex r g) k x = true) - <-> - (VertexSet.In x (interference_adj r g) /\ interf_degree g x = k)). - -Proof. -intros. split; intros; destruct H0. -split. eapply low_degree_in_interf; eauto. eapply degree_dec_remove_K; eauto. -cut (has_low_degree g k x = false). intro. -split. assumption. -apply degree_K_remove_dec; auto. -unfold has_low_degree. rewrite H1. destruct (le_lt_dec k k). auto. elim (lt_irrefl _ l). -Qed. diff --git a/backend/Remove_Vertex_Move.v b/backend/Remove_Vertex_Move.v deleted file mode 100755 index a473473..0000000 --- a/backend/Remove_Vertex_Move.v +++ /dev/null @@ -1,206 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import WS. -Require Import Edges. -Require Import MyRegisters. -Require Import Affinity_relation. -Require Import Interference_adjacency. - -Module Register := Regs. - -Import RegFacts Props Edge. - -(* A nonmove-related vertex of g is not move-related - after the removal of a vertex *) -Lemma move_remove : forall x r g, -move_related g x = false -> -move_related (remove_vertex r g) x = false. - -Proof. -intros. -apply move_related_false_charac2. intros. -apply move_related_false_charac with (g:=g); auto. -rewrite In_remove_edge in H1. intuition. -Qed. - -(* Equivalently, a move-related vertex of (remove_vertex r g) - is move-related in g *) -Lemma move_remove_2 : forall x r g, -move_related (remove_vertex r g) x = true -> -move_related g x = true. - -Proof. -intros x r g H. -generalize (move_related_charac _ _ H);intro H0. -destruct H0 as [y H0]. destruct H0 as [H0 H1]. destruct H1 as [H1 H2]. -apply move_related_charac2 with (e:= y). -assumption. -rewrite In_remove_edge in H1. destruct H1. assumption. assumption. -Qed. - -(* If x (different from r) is move-related in g and nonmove-related - in (remove_vertex r g) then x is a preference neighbor of r in g *) -Lemma move_related_not_remove_in_pref : forall x r g, -~Register.eq r x -> -move_related g x = true -> -move_related (remove_vertex r g) x = false -> -VertexSet.In x (preference_adj r g). - -Proof. -intros. -generalize (move_related_charac _ _ H0). intro. -destruct H2. destruct H2. destruct H3. -destruct (incident_dec x0 r). -destruct H5. destruct H4. -elim H. rewrite H4. rewrite H5. auto. -unfold aff_edge in H2. destruct H2. -rewrite in_pref. exists x1. -assert (eq (x,r,Some x1) x0). -rewrite edge_comm. apply eq_ordered_eq. -unfold E.eq; split; simpl. split; auto. -unfold get_weight in H2. rewrite H2. apply OptionN_as_OT.eq_refl. -rewrite H6. assumption. -destruct H4. -destruct H2. -rewrite in_pref. exists x1. -assert (eq (x,r,Some x1) x0). -apply eq_ordered_eq. -unfold E.eq; split; simpl. split; auto. -unfold get_weight in H2. rewrite H2. apply OptionN_as_OT.eq_refl. -rewrite H6. assumption. -elim H. rewrite H4. rewrite H5. auto. -assert (In_graph_edge x0 (remove_vertex r g)). -rewrite In_remove_edge. split; assumption. -generalize (Aff_edge_aff _ _ H6 H2). intro. -destruct H7. destruct H4. -rewrite (compat_bool_move _ _ _ (Register.eq_sym _ _ H4)) in H7. rewrite H1 in H7. inversion H7. -rewrite (compat_bool_move _ _ _ (Register.eq_sym _ _ H4)) in H8. rewrite H1 in H8. inversion H8. -Qed. - -(* The preference neighborhood of any vertex x different from r in - (remove_vertex r g) is obtained by removing r from the interference - neighborhood of x in g *) -Lemma remove_pref_adj : forall x r g, -~Register.eq x r -> -VertexSet.Equal (preference_adj x (remove_vertex r g)) - (VertexSet.remove r (preference_adj x g)). - -Proof. -intros. -split; intros. -apply VertexSet.remove_2. intro. -rewrite <-H1 in H0. rewrite in_pref in H0. destruct H0. -generalize (proj1 (In_graph_edge_in_ext _ _ H0)). change_rewrite. intro. -rewrite In_remove_vertex in H2. destruct H2. elim (H3 (Register.eq_refl _)). -rewrite in_pref. rewrite in_pref in H0. destruct H0. exists x0. -rewrite In_remove_edge in H0. destruct H0. assumption. - -generalize (VertexSet.remove_3 H0). intro. -rewrite in_pref in H1. destruct H1. -rewrite in_pref. exists x0. rewrite In_remove_edge. split. assumption. -intro. destruct H2; change_rewrite. -elim (VertexSet.remove_1 H2 H0). -elim H. intuition. -Qed. - -(* The preference degree of any vertex which is move-related - in g and nonmove-related in (remove_vertex r g) is equal to 1 *) -Lemma pref_degree_dec_remove_1 : forall x r g, -~Register.eq x r -> -move_related g x = true -> -move_related (remove_vertex r g) x = false -> -pref_degree g x = 1. - -Proof. -unfold pref_degree. intros. -generalize (not_move_related_empty_pref _ _ H1). intro. -generalize (remove_pref_adj x r g H). intro. -rewrite H2 in H3. -cut (~Register.eq r x). intro. -generalize (move_related_not_remove_in_pref _ _ _ H4 H0 H1). intro. -generalize (pref_adj_comm _ _ _ H5). intro. -generalize (remove_cardinal_1 H6). intro. -rewrite <-H7. apply eq_S. rewrite <-H3. -rewrite <-Props.cardinal_Empty. apply VertexSet.empty_1. -intuition. -Qed. - -Lemma cardinal_1_singleton : forall x s, -VertexSet.In x s -> -VertexSet.cardinal s = 1 -> -VertexSet.Equal s (VertexSet.singleton x). - -Proof. -intros. -apply VertexSet.eq_sym. -apply remove_singleton_empty. assumption. -assert (VertexSet.cardinal (VertexSet.remove x s) = 0). -rewrite <-remove_cardinal_1 with (x:=x) in H0. auto. -assumption. -rewrite <-Props.cardinal_Empty in H1. -rewrite (empty_is_empty_1 H1). apply VertexSet.eq_refl. -Qed. - -(* Reciprocally, any vertex different from r which has a preference - degree equal to 1 in g and is a preference neighbor of r in g is - nonmove-related in (remove_vertex r g) *) -Lemma pref_degree_1_remove_dec : forall x r g, -~Register.eq x r -> -pref_degree g x = 1 -> -VertexSet.In x (preference_adj r g) -> -move_related (remove_vertex r g) x = false. - -Proof. -intros. -case_eq (move_related (remove_vertex r g) x); intros. -generalize (remove_pref_adj x r g H). intro. -assert (VertexSet.Equal (preference_adj x g) (VertexSet.singleton r)). -apply cardinal_1_singleton. apply pref_adj_comm. assumption. assumption. -rewrite H4 in H3. -cut (VertexSet.Equal (preference_adj x (remove_vertex r g)) VertexSet.empty). intro. -generalize (move_related_charac _ _ H2). intro. -destruct H6. destruct H6. destruct H7. -destruct H8. -assert (VertexSet.In (snd_ext x0) (preference_adj x (remove_vertex r g))). -destruct H6. rewrite in_pref. exists x1. -assert (eq (snd_ext x0, x, Some x1) x0). -rewrite edge_comm. apply eq_ordered_eq. -unfold E.eq. split; intros. simpl. split; intuition. apply Regs.eq_refl. -auto. auto. simpl. unfold get_weight in H6. rewrite H6. apply OptionN_as_OT.eq_refl. -rewrite H9. assumption. -rewrite H5 in H9. elim (VertexSet.empty_1 H9). -assert (VertexSet.In (fst_ext x0) (preference_adj x (remove_vertex r g))). -destruct H6. rewrite in_pref. exists x1. -assert (eq (fst_ext x0, x, Some x1) x0). -apply eq_ordered_eq. -unfold E.eq. split; intros. simpl. split; intuition. apply Regs.eq_refl. -auto. auto. simpl. unfold get_weight in H6. rewrite H6. apply OptionN_as_OT.eq_refl. -rewrite H9. assumption. -rewrite H5 in H9. elim (VertexSet.empty_1 H9). -rewrite H3. -split; intros. -destruct (Register.eq_dec r a). -elim (VertexSet.remove_1 e H5). -generalize (VertexSet.remove_3 H5). intro. -generalize (VertexSet.singleton_1 H6). intro. -elim (n H7). -elim (VertexSet.empty_1 H5). -reflexivity. -Qed. - -(* Meaningful theorem *) -Theorem Remove_vertex_move_evolution : forall x r g, -~Register.eq x r -> -((move_related g x = true /\ move_related (remove_vertex r g) x = false) - <-> - (pref_degree g x = 1 /\ VertexSet.In x (preference_adj r g))). - -Proof. -split; intros. -destruct H0. -split. apply pref_degree_dec_remove_1 with (r:=r); auto. - apply move_related_not_remove_in_pref; auto. -destruct H0. -split. apply move_related_card. congruence. - apply pref_degree_1_remove_dec; auto. -Qed. diff --git a/backend/Remove_Vertex_WL.v b/backend/Remove_Vertex_WL.v deleted file mode 100755 index 2ad7c76..0000000 --- a/backend/Remove_Vertex_WL.v +++ /dev/null @@ -1,478 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import IRC_graph. -Require Import ZArith. -Require Import Edges. -Require Import Remove_Vertex_Degree. -Require Import WS. -Require Import Remove_Vertex_Move. -Require Import Affinity_relation. -Require Import Interference_adjacency. - -Import RegFacts Props OTFacts. - -Definition eq_K x K := match eq_nat_dec x K with -|left _ => true -|right _ => false -end. - -Lemma eq_K_1 : forall x y, -y = x -> -eq_K x y = true. - -Proof. -intros. unfold eq_K. rewrite <-H. destruct (eq_nat_dec y y); intuition. -Qed. - -Lemma eq_K_2 : forall x y, -eq_K x y = true -> -x = y. - -Proof. -intros. unfold eq_K in H. destruct (eq_nat_dec x y). auto. inversion H. -Qed. - -Lemma eq_K_compat : forall K g, -compat_bool Register.eq (fun x => eq_K K (VertexSet.cardinal (interference_adj x g))). - -Proof. -unfold compat_bool. intros. rewrite (compat_interference_adj _ _ _ H). reflexivity. -Qed. - -Lemma eq_K_compat_pref : forall K g, -compat_bool Register.eq (fun x => eq_K K (VertexSet.cardinal (preference_adj x g))). - -Proof. -unfold compat_bool. intros. rewrite (compat_preference_adj _ _ _ H). reflexivity. -Qed. - -Lemma compat_move_up : forall g K, -compat_bool Register.eq -(fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g K x). - -Proof. -unfold compat_bool; intros. -rewrite (compat_preference_adj _ _ _ H). -destruct (eq_K 1 (VertexSet.cardinal (preference_adj y g))). simpl. -apply compat_bool_low. assumption. -simpl. reflexivity. -Qed. - -Definition remove_wl_2 r ircg K := - let g := irc_g ircg in - let wl := irc_wl ircg in - let simplify := get_simplifyWL wl in - let freeze := get_freezeWL wl in - let spillWL := get_spillWL wl in - let movesWL := get_movesWL wl in - let pre := precolored g in - let int_adj := interference_adj r g in - let not_pre_int_adj := VertexSet.diff int_adj pre in - let pre_adj := preference_adj r g in - let not_pre_pre_adj := VertexSet.diff pre_adj pre in - let newlow := VertexSet.filter (fun x => eq_K K (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj in - let (free, simp) := VertexSet.partition (move_related g) newlow in - let newnmr := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g K x) not_pre_pre_adj in - let simplifyWL__ := VertexSet.union simplify simp in - let simplifyWL_ := VertexSet.union simplifyWL__ newnmr in - let simplifyWL' := VertexSet.remove r simplifyWL_ in - let freezeWL__ := VertexSet.diff freeze newnmr in - let freezeWL_ := VertexSet.union freezeWL__ free in - let freezeWL' := VertexSet.remove r freezeWL_ in - let spillWL_ := VertexSet.diff spillWL newlow in - let spillWL' := VertexSet.remove r spillWL_ in - let movesWL' := not_incident_edges r movesWL g in - (spillWL', freezeWL', simplifyWL', movesWL'). - -Lemma WS_remove_wl_2 : forall r ircg, -WS_properties (remove_vertex r (irc_g ircg)) (VertexSet.cardinal (pal ircg)) - (remove_wl_2 r ircg (VertexSet.cardinal (pal ircg))). - -Proof. -unfold remove_wl_2. intros r ircg. -generalize (HWS_irc ircg). intro HWS. rewrite <-(Hk ircg) in *. -set (g' := remove_vertex r (irc_g ircg)) in *. -set (k := VertexSet.cardinal (pal ircg)) in *. -set (g := irc_g ircg) in *. -set (wl := irc_wl ircg) in *. -set ( simplify := get_simplifyWL wl ) in *. -set ( freeze := get_freezeWL wl ) in *. -set ( spillWL := get_spillWL wl ) in *. -set ( int_adj := interference_adj r g ) in *. -set ( not_pre_int_adj := VertexSet.diff int_adj (precolored g) ) in *. -set ( pre_adj := preference_adj r g ) in *. -set ( not_pre_pre_adj := VertexSet.diff pre_adj (precolored g) ) in *. -set ( low := VertexSet.filter (fun x => eq_K k (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj ) in *. -set ( simpfree := VertexSet.partition (move_related g) low ) in *. -case_eq simpfree. intros free simp Hsf. -unfold simpfree in Hsf. -set ( nmr := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g k x) not_pre_pre_adj) in *. -set ( simplifyWL__ := VertexSet.union simplify simp ) in *. -set ( simplifyWL_ := VertexSet.union simplifyWL__ nmr) in *. -set ( simplifyWL' := VertexSet.remove r simplifyWL_ ) in *. -set ( freezeWL__ := VertexSet.diff freeze nmr ) in *. -set ( freezeWL_ := VertexSet.union freezeWL__ free ) in *. -set ( freezeWL' := VertexSet.remove r freezeWL_ ) in *. -set ( spillWL_ := VertexSet.diff spillWL low ) in *. -set ( spillWL' := VertexSet.remove r spillWL_ ) in *. -set ( movesWL' := not_incident_edges r (get_movesWL wl) g) in *. - -unfold WS_properties. split. -split; intro H. -unfold get_spillWL in H. simpl in H. -(* spillWL' => *) -unfold spillWL' in H. -generalize (VertexSet.remove_3 H). intro H0. -unfold spillWL_ in H0. -generalize (VertexSet.diff_1 H0). intro H1. -generalize (VertexSet.diff_2 H0). intro H2. -split. -case_eq (has_low_degree g' k x); intros. -elim H2. apply VertexSet.filter_3. -apply eq_K_compat. -unfold not_pre_int_adj. apply VertexSet.diff_3. -apply low_degree_in_interf with (K := k). -assumption. -intro. elim (VertexSet.remove_1 (Register.eq_sym _ _ H4) H). -apply (proj1 (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS)). -apply (proj2 (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS)). -apply eq_K_1. -apply degree_dec_remove_K with (r := r). -intro. elim (VertexSet.remove_1 (Register.eq_sym _ _ H4) H). -apply (proj1 (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS)). -assumption. -reflexivity. -split. -unfold g'. rewrite In_remove_vertex. split. -apply (proj2 (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS)). -intro. elim (VertexSet.remove_1 (Register.eq_sym _ _ H3) H). -unfold g'. rewrite precolored_remove_vertex. -intro. elim (proj2 (proj2 (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). -apply (VertexSet.remove_3 H3). - -(* spillWL' <= *) -destruct H. destruct H0. -unfold get_spillWL. simpl. -unfold spillWL'. -assert (~Register.eq r x) as Hr. -intro. rewrite <-H2 in H0. unfold g' in H0. -rewrite In_remove_vertex in H0. destruct H0. elim H3. auto. -apply VertexSet.remove_2. assumption. -unfold spillWL_. apply VertexSet.diff_3. -WS_apply HWS. -split. -apply not_low_remove_not_low with (r:=r). -assumption. -intuition. -split. -unfold g' in H0. rewrite In_remove_vertex in H0. intuition. -unfold g' in H1. rewrite precolored_remove_vertex in H1. -intro. elim H1. apply VertexSet.remove_2; auto. -intro. unfold low in H2. -generalize (VertexSet.filter_1 (eq_K_compat k g) H2). intro H3. -generalize (VertexSet.filter_2 (eq_K_compat k g) H2). clear H2. intro H2. -generalize (eq_K_2 _ _ H2). clear H2. intro H2. -assert (has_low_degree g' k x = true). -apply degree_K_remove_dec. intuition. -unfold has_low_degree, interf_degree. rewrite <-H2. -destruct (le_lt_dec k k). reflexivity. elim (lt_irrefl _ l). -auto. -unfold not_pre_int_adj in H3. generalize (VertexSet.diff_1 H3). intro. -unfold int_adj in H4. assumption. -rewrite H in H4. inversion H4. - -(* freezeWL' => *) -split. -unfold get_freezeWL. simpl. split; intros. -unfold freezeWL' in H. -assert (~Register.eq r x) as Hr. -intro. elim (VertexSet.remove_1 H0 H). -generalize (VertexSet.remove_3 H). clear H. intro. -unfold freezeWL_ in H. -destruct (VertexSet.union_1 H). -unfold freezeWL__ in H0. -generalize (VertexSet.diff_1 H0). intro. -generalize (VertexSet.diff_2 H0). clear H0. intro. -split. -apply low_remove_low. -apply (proj1 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS)). -intuition. -split. -unfold nmr in H0. -case_eq (move_related g' x); intros. -reflexivity. -elim H0. apply VertexSet.filter_3. -apply compat_move_up. -unfold not_pre_pre_adj. apply VertexSet.diff_3. -unfold pre_adj. apply move_related_not_remove_in_pref. intuition. -apply (proj1 (proj2 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). -assumption. -apply (proj2 (proj2 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). -assert (VertexSet.cardinal (preference_adj x g) = 1). -apply pref_degree_dec_remove_1 with (r:=r). intuition. -apply (proj1 (proj2 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). assumption. -rewrite (eq_K_1 _ _ H3). simpl. -apply (proj1 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS)). -unfold g'. rewrite precolored_remove_vertex. -intro. elim (proj2 (proj2 (proj2 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS)))). -apply (VertexSet.remove_3 H2). -split. -assert (VertexSet.cardinal (interference_adj x g) = k). -assert (VertexSet.In x (fst (VertexSet.partition (move_related g) low))). -rewrite Hsf. simpl. assumption. -rewrite (VertexSet.partition_1) in H1. -generalize (VertexSet.filter_1 (compat_bool_move _) H1). intro. -unfold low in H2. -generalize (VertexSet.filter_1 (eq_K_compat k g) H2). intro H3. -generalize (VertexSet.filter_2 (eq_K_compat k g) H2). clear H2. intro H2. -generalize (eq_K_2 _ _ H2). auto. apply compat_bool_move. -apply degree_K_remove_dec. -intuition. -unfold has_low_degree, interf_degree. rewrite H1. -destruct (le_lt_dec k k). reflexivity. elim (lt_irrefl _ l). auto. -assert (VertexSet.In x (fst (VertexSet.partition (move_related g) low))). -rewrite Hsf. simpl. assumption. -rewrite (VertexSet.partition_1) in H2. -generalize (VertexSet.filter_1 (compat_bool_move _) H2). intro. -unfold low in H3. -generalize (VertexSet.filter_1 (eq_K_compat k g) H3). intro H4. -generalize (VertexSet.filter_2 (eq_K_compat k g) H3). clear H3. intro H3. -unfold not_pre_int_adj in H4. apply (VertexSet.diff_1 H4). -apply compat_bool_move. -split. -case_eq (move_related g' x); intros. -reflexivity. -assert (VertexSet.In x (fst (VertexSet.partition (move_related g) low))). -rewrite Hsf. simpl. assumption. -rewrite (VertexSet.partition_1) in H2. -generalize (VertexSet.filter_1 (compat_bool_move _) H2). intro. -generalize (VertexSet.filter_2 (compat_bool_move _) H2). clear H2. intro H2. -assert (VertexSet.In x (preference_adj r g)). -apply move_related_not_remove_in_pref; assumption. -assert (VertexSet.In x (interference_adj r g)). -unfold low in H3. -generalize (VertexSet.filter_1 (eq_K_compat k g) H3). intro H5. -apply (VertexSet.diff_1 H5). -elim (interf_pref_conflict x r g). split. -rewrite in_pref in H4. destruct H4. -unfold Prefere. exists x0. assumption. -rewrite in_interf in H5. unfold Interfere. assumption. -apply compat_bool_move. -unfold g'. rewrite precolored_remove_vertex. -intro. -assert (VertexSet.In x (fst (VertexSet.partition (move_related g) low))). -rewrite Hsf. simpl. assumption. -rewrite (VertexSet.partition_1) in H2. -generalize (VertexSet.filter_1 (compat_bool_move _) H2). intro. -unfold low in H3. -generalize (VertexSet.filter_1 (eq_K_compat k g) H3). intro. -elim (VertexSet.diff_2 H4). apply (VertexSet.remove_3 H1). -apply compat_bool_move. - -(* freezeWL' <= *) -destruct H. destruct H0. -unfold freezeWL'. -assert (~Register.eq r x). -intro. generalize (move_related_in_graph _ _ H0). intro. rewrite <-H2 in H3. -unfold g' in H3. rewrite In_remove_vertex in H3. destruct H3. elim H4. auto. -apply VertexSet.remove_2. assumption. -unfold freezeWL_. -case_eq (has_low_degree g k x); intros. -apply VertexSet.union_2. unfold freezeWL__. -apply VertexSet.diff_3. -WS_apply HWS. -split. -assumption. -split. -apply move_remove_2 with (r:=r). assumption. -unfold g' in H1. rewrite precolored_remove_vertex in H1. -intro. elim H1. apply VertexSet.remove_2. assumption. assumption. -intro. unfold nmr in H4. -assert (move_related g' x = false). -apply pref_degree_1_remove_dec. -intuition. -generalize (VertexSet.filter_2 (compat_move_up g k) H4). intro. -case_eq (eq_K 1 (VertexSet.cardinal (preference_adj x g))); intros. -rewrite (eq_K_2 _ _ H6); auto. -rewrite H6 in H5. inversion H5. -generalize (VertexSet.filter_1 (compat_move_up g k) H4). intro. -apply (VertexSet.diff_1 H5). -rewrite H0 in H5. inversion H5. -apply VertexSet.union_3. -cut (VertexSet.In x (fst (VertexSet.partition (move_related g) low))). intro. -rewrite Hsf in H4. simpl in H4. assumption. -rewrite VertexSet.partition_1. -apply VertexSet.filter_3. -apply compat_bool_move. -unfold low. apply VertexSet.filter_3. -apply eq_K_compat. -apply VertexSet.diff_3. -unfold int_adj. -apply low_degree_in_interf with (K:=k). -assumption. intuition. assumption. -unfold g' in H1. rewrite precolored_remove_vertex in H1. -intro. elim H1. apply VertexSet.remove_2; auto. -apply eq_K_1. -apply degree_dec_remove_K with (r:=r). -intuition. assumption. assumption. -apply move_remove_2 with (r:=r). assumption. -apply compat_bool_move. - -(* simplifyWL' => *) -split. -unfold get_simplifyWL. simpl. -split; intros. -unfold simplifyWL' in H. -assert (~Register.eq r x) as Hr. -intro. elim (VertexSet.remove_1 H0 H). -generalize (VertexSet.remove_3 H). clear H. intro H. -unfold simplifyWL_ in H. -destruct (VertexSet.union_1 H). -unfold simplifyWL__ in H0. -destruct (VertexSet.union_1 H0). -generalize (In_simplify_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS). clear H1. intro. -destruct H1. destruct H2. destruct H3. -split. apply low_remove_low. assumption. intuition. -split. unfold g'. -apply move_remove. assumption. -split. unfold g'. rewrite In_remove_vertex. split; auto. -unfold g'. rewrite precolored_remove_vertex. intro. -elim H4. apply (VertexSet.remove_3 H5). -assert (VertexSet.In x (snd (VertexSet.partition (move_related g) low))). -rewrite Hsf. simpl. assumption. -rewrite VertexSet.partition_2 in H2. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_move g)) H2). intro. -generalize (VertexSet.filter_2 (compat_not_compat (compat_bool_move g)) H2). clear H2. intro. -unfold low in H3. -generalize (VertexSet.filter_1 (eq_K_compat k g) H3). intro. -generalize (VertexSet.filter_2 (eq_K_compat k g) H3). clear H3. intro. -generalize (eq_K_2 _ _ H3). clear H3. intro. -split. -apply degree_K_remove_dec. intuition. -unfold has_low_degree, interf_degree. rewrite <-H3. -destruct (le_lt_dec k k). auto. elim (lt_irrefl _ l). auto. -apply (VertexSet.diff_1 H4). -split. -apply move_remove. -destruct (move_related g x). inversion H2. reflexivity. -split. -unfold g'. rewrite In_remove_vertex. split. -apply in_interf_in with (r:=r). -apply (VertexSet.diff_1 H4). intuition. -unfold g'. rewrite precolored_remove_vertex. intro. -elim (VertexSet.diff_2 H4). apply (VertexSet.remove_3 H5). -apply compat_bool_move. -(* x in nmr *) -unfold nmr in H0. -generalize (VertexSet.filter_1 (compat_move_up g k) H0). intro. -generalize (VertexSet.filter_2 (compat_move_up g k) H0). clear H0. intro. -assert (VertexSet.cardinal (preference_adj x g) = 1). -symmetry. apply eq_K_2. -case_eq (eq_K 1 (VertexSet.cardinal (preference_adj x g))); intros. -reflexivity. -rewrite H2 in H0. inversion H0. -rewrite (eq_K_1 _ _ H2) in H0. simpl in H0. -split. -apply low_remove_low. assumption. intuition. -split. -case_eq (move_related g x); intros. -apply pref_degree_1_remove_dec. intuition. assumption. -apply (VertexSet.diff_1 H1). -apply move_remove. assumption. -split. unfold g'. rewrite In_remove_vertex. split. apply in_pref_in with (r:=r). -apply (VertexSet.diff_1 H1). intuition. -unfold g'. rewrite precolored_remove_vertex. intro. -elim (VertexSet.diff_2 H1). apply (VertexSet.remove_3 H3). - -(* simplifyWL' <= *) -destruct H. destruct H0. destruct H1. -unfold simplifyWL'. -assert (~Register.eq r x) as Hr. -intro. rewrite <-H3 in H1. -unfold g' in H1. rewrite In_remove_vertex in H1. destruct H1. elim H4. auto. -apply VertexSet.remove_2. assumption. -unfold simplifyWL_. -case_eq (has_low_degree g k x); intros. -case_eq (move_related g x); intros. -apply VertexSet.union_3. -unfold nmr. apply VertexSet.filter_3. -apply compat_move_up. -apply VertexSet.diff_3. -apply move_related_not_remove_in_pref. assumption. assumption. assumption. -unfold g' in H2. rewrite precolored_remove_vertex in H2. -intro. elim H2. apply VertexSet.remove_2. intuition. assumption. -assert (eq_K 1 (VertexSet.cardinal (preference_adj x g)) = true). -apply eq_K_1. -apply pref_degree_dec_remove_1 with (r:=r). intuition . assumption. assumption. -rewrite H5. simpl. assumption. -apply VertexSet.union_2. -unfold simplifyWL__. -apply VertexSet.union_2. -WS_apply HWS. -split. -assumption. -split. -assumption. -split. -unfold g' in H1. rewrite In_remove_vertex in H1. intuition. -unfold g' in H2. rewrite precolored_remove_vertex in H2. -intro. elim H2. apply VertexSet.remove_2. intuition. assumption. -apply VertexSet.union_2. -unfold simplifyWL__. -apply VertexSet.union_3. -assert (VertexSet.In x (snd (VertexSet.partition (move_related g) low))). -rewrite VertexSet.partition_2. -apply VertexSet.filter_3. -apply compat_not_compat. apply compat_bool_move. -unfold low. -apply VertexSet.filter_3. -apply eq_K_compat. apply VertexSet.diff_3. -apply low_degree_in_interf with (K:=k). assumption. intuition. assumption. -unfold g' in H2. rewrite precolored_remove_vertex in H2. -intro. elim H2. apply VertexSet.remove_2. intuition. assumption. -apply eq_K_1. apply degree_dec_remove_K with (r:=r). intuition. assumption. assumption. -case_eq (move_related g x); intros. -assert (VertexSet.In x (interference_adj r g)). -apply low_degree_in_interf with (K:=k). assumption. intuition. assumption. -assert (VertexSet.In x (preference_adj r g)). -apply move_related_not_remove_in_pref. assumption. assumption. assumption. -elim (interf_pref_conflict x r g). -split. -unfold Prefere. rewrite in_pref in H6. assumption. -unfold Interfere. rewrite <-in_interf. assumption. -auto. -apply compat_bool_move. -rewrite Hsf in H4. simpl in H4. assumption. - -(* movesWL => *) -unfold movesWL', get_movesWL. simpl. -split; intros. -simpl in H. -rewrite not_incident_edges_1 in H. destruct H as [H HH]. -generalize (In_move_props _ _ _ _ _ _ _ _ H (refl_equal _) HWS). clear H. intro H. -destruct H. destruct H0. -split. assumption. -unfold g'. rewrite In_remove_edge. split. -apply (proj2 (proj1 (In_graph_aff_edge_in_AE _ _) H0)). -assumption. -generalize (proj1 (In_graph_interf_edge_in_IE _ _) H0). intro. -destruct H1. -inversion H. rewrite H1 in H3. inversion H3. -intros. apply (In_move_props _ _ _ _ _ _ _ _ H0 (refl_equal _) HWS). -destruct H. -rewrite not_incident_edges_1. -split. -WS_apply HWS. -split. -assumption. -unfold g' in H0. rewrite In_remove_edge in H0. intuition. -intro H1. -elim (In_graph_edge_in_ext _ _ H0). -intros. -destruct H1. -rewrite <-H1 in H2. -unfold g' in H2. rewrite In_remove_vertex in H2. destruct H2. elim H4. auto. -rewrite <-H1 in H3. -unfold g' in H3. rewrite In_remove_vertex in H3. destruct H3. elim H4. auto. -intros. apply (In_move_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS). -Qed. diff --git a/backend/SetsFacts.v b/backend/SetsFacts.v deleted file mode 100755 index 60145cf..0000000 --- a/backend/SetsFacts.v +++ /dev/null @@ -1,105 +0,0 @@ -Require Import FSets. -Set Implicit Arguments. - -(* Useful properties over finite sets and ordered types *) - -Module MyOTFacts (M : OrderedType). - -Section compat. - -Lemma compat_not_compat : forall f : M.t -> bool, -compat_bool M.eq f -> -compat_bool M.eq (fun x => negb (f x)). - -Proof. -unfold compat_bool;intros f H. -intros; unfold negb. -rewrite (H x y H0);reflexivity. -Qed. - -End compat. - -End MyOTFacts. - -Module MyFacts (M : S). - -Import M. -Module Import Props := Properties M. -Module Import Facts := OrderedTypeFacts E. -Module Import OTFacts := MyOTFacts E. - -Lemma set_induction2 : forall s, -Empty s \/ exists x, exists s', Add x s' s. - -Proof. -intros. case_eq (choose s);intros. -right. exists e. exists (remove e s). -constructor;intro H0. -destruct (eq_dec e y). -left;assumption. -right;apply (remove_2 n H0). -destruct H0. -rewrite <-H0;apply (choose_1 H). -eapply remove_3;eassumption. -left;apply (choose_2 H). -Qed. - -Lemma equal_equivlist : forall s s', -Equal s s' -> equivlistA E.eq (elements s) (elements s'). - -Proof. -unfold equivlistA. -generalize elements_1;generalize elements_2;intros H0 H1 s s' H x. -split;intro H2. -apply H1;rewrite <-H;apply H0;assumption. -apply H1;rewrite H;apply H0;assumption. -Qed. - -Section Fold_Facts. - -Variable A : Type. - -Lemma fold_left_compat_set : forall (f : t -> A -> t) l e e', -Equal e e' -> -(forall e1 e2 a, Equal e1 e2 -> Equal (f e1 a) (f e2 a)) -> -Equal (fold_left f l e) (fold_left f l e'). - -Proof. -intros f l. -induction l;simpl. -auto. -intros e e' H H0 H1. -apply (IHl (f e a) (f e' a)). -apply H0;assumption. -assumption. -Qed. - -Lemma fold_left_assoc : forall l f x h, -(forall (y z : A) s, Equal (f (f s y) z) (f (f s z) y)) -> -(forall e1 e2 a, Equal e1 e2 -> Equal (f e1 a) (f e2 a)) -> -Equal (fold_left f (h :: l) x) (f (fold_left f l x) h). - -Proof. -induction l;simpl;intros f x h H H0. -intuition. -rewrite <-IHl;simpl;try assumption. -apply fold_left_compat_set;[apply H|];auto. -Qed. - -Lemma NoDupA_elements : forall s, -NoDupA E.eq (elements s). - -Proof. -intro s. -apply SortA_NoDupA with (ltA := E.lt). -apply E.eq_refl. -apply E.eq_sym. -apply E.lt_trans. -apply E.lt_not_eq. -apply Facts.lt_eq. -apply Facts.eq_lt. -apply elements_3. -Qed. - -End Fold_Facts. -End MyFacts.
\ No newline at end of file diff --git a/backend/Simplify_WL.v b/backend/Simplify_WL.v deleted file mode 100755 index 4da207a..0000000 --- a/backend/Simplify_WL.v +++ /dev/null @@ -1,160 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Remove_Vertex_WL. -Require Import WS. -Require Import Affinity_relation. -Require Import Interference_adjacency. -Require Import Edges. -Require Import IRC_graph. - -Import Edge RegFacts Props OTFacts. - -Definition simplify_wl r ircg K := - let g := irc_g ircg in - let wl := irc_wl ircg in - let simplify := get_simplifyWL wl in - let freeze := get_freezeWL wl in - let spillWL := get_spillWL wl in - let movesWL := get_movesWL wl in - let pre := precolored g in - let int_adj := interference_adj r g in - let not_pre_int_adj := VertexSet.diff int_adj pre in - let newlow := VertexSet.filter (fun x => eq_K K (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj in - let (free, simp) := VertexSet.partition (move_related g) newlow in - let simplifyWL_ := VertexSet.union simplify simp in - let simplifyWL' := VertexSet.remove r simplifyWL_ in - let freezeWL' := VertexSet.union freeze free in - let spillWL' := VertexSet.diff spillWL newlow in - let movesWL' := movesWL in (spillWL', freezeWL', simplifyWL', movesWL'). - -Lemma WS_simplify_aux : forall r ircg, -VertexSet.In r (get_simplifyWL (irc_wl ircg)) -> -WS_properties (remove_vertex r (irc_g ircg)) (VertexSet.cardinal (pal ircg)) - (simplify_wl r ircg (VertexSet.cardinal (pal ircg))). - -Proof. -intros. -generalize (WS_props_equal (remove_vertex r (irc_g ircg)) (VertexSet.cardinal (pal ircg)) - (remove_wl_2 r ircg (VertexSet.cardinal (pal ircg))) - (simplify_wl r ircg (VertexSet.cardinal (pal ircg)))). -generalize (WS_remove_wl_2 r ircg). -unfold remove_wl_2, simplify_wl, get_simplifyWL, - get_freezeWL, get_spillWL, get_movesWL. -set (g' := remove_vertex r (irc_g ircg)) in *. -set (k := VertexSet.cardinal (pal ircg)) in *. -set (g := irc_g ircg) in *. -set (wl := irc_wl ircg) in *. -set ( simplify := get_simplifyWL wl ) in *. -set ( freeze := get_freezeWL wl ) in *. -set ( spillWL := get_spillWL wl ) in *. -set ( int_adj := interference_adj r g ) in *. -set ( not_pre_int_adj := VertexSet.diff int_adj (precolored g) ) in *. -set ( pre_adj := preference_adj r g ) in *. -set ( not_pre_pre_adj := VertexSet.diff pre_adj (precolored g) ) in *. -set ( low := VertexSet.filter (fun x => eq_K k (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj ) in *. -set ( simpfree := VertexSet.partition (move_related g) low ) in *. -case_eq simpfree. intros free simp Hsf. -unfold simpfree in Hsf. -set ( nmr := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g k x) not_pre_pre_adj) in *. -set ( simplifyWL__ := VertexSet.union simplify simp ) in *. -set ( simplifyWL_ := VertexSet.union simplifyWL__ nmr) in *. -set ( simplifyWL' := VertexSet.remove r simplifyWL_ ) in *. -set ( freezeWL__ := VertexSet.diff freeze nmr ) in *. -set ( freezeWL_ := VertexSet.union freezeWL__ free ) in *. -set ( freezeWL' := VertexSet.remove r freezeWL_ ) in *. -set ( spillWL_ := VertexSet.diff spillWL low ) in *. -set ( spillWL' := VertexSet.remove r spillWL_ ) in *. -set ( movesWL' := not_incident_edges r (get_movesWL wl) g) in *. simpl. -generalize (In_simplify_props _ _ _ _ _ _ _ _ H (refl_equal _) (HWS_irc ircg)). intro Hr. - -assert (VertexSet.Equal (preference_adj r g) VertexSet.empty) as Hempty_pre. -apply not_move_related_empty_pref. intuition. -assert (VertexSet.Equal nmr VertexSet.empty) as Hempty_nmr. -unfold nmr. -split; intros. -generalize (VertexSet.filter_1 (compat_move_up _ _) H0). intro. -unfold not_pre_pre_adj in H1. generalize (VertexSet.diff_1 H1). intro. -unfold pre_adj in H2. rewrite Hempty_pre in H2. elim (VertexSet.empty_1 H2). -elim (VertexSet.empty_1 H0). - -intros HWS H0. apply H0. - -(* simplify worklist is only decreasing from r and increasing from simp, - since nmr is empty, because preference adj r g is empty *) -split; intros. -apply VertexSet.remove_2. -intro. elim (VertexSet.remove_1 H2 H1). -destruct (VertexSet.union_1 (VertexSet.remove_3 H1)). -assumption. -rewrite Hempty_nmr in H2. elim (VertexSet.empty_1 H2). - -apply VertexSet.remove_2. -intro. elim (VertexSet.remove_1 H2 H1). -apply VertexSet.union_2. apply (VertexSet.remove_3 H1). - -(* r is not deleted from freezewl, since it is not move-related and - no vertex is removed from freeze, since free is empty, because - preference adj r g is empty *) - -set (s := VertexSet.union (VertexSet.diff (snd (fst (fst wl))) nmr) free). -split; intros. -unfold s in H1. destruct (VertexSet.union_1 (VertexSet.remove_3 H1)). -apply VertexSet.union_2. apply (VertexSet.diff_1 H2). -apply VertexSet.union_3. assumption. - -apply VertexSet.remove_2. intro. rewrite <-H2 in H1. clear H2. -unfold s in H1. -destruct (VertexSet.union_1 H1). -generalize (In_freeze_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). intro. -destruct H3. destruct H4. destruct Hr. destruct H7. -rewrite H7 in H4. inversion H4. -assert (free = fst (VertexSet.partition (move_related g) low)). -rewrite Hsf. auto. -rewrite H3 in H2. rewrite VertexSet.partition_1 in H2. -generalize (VertexSet.filter_2 (compat_bool_move _ ) H2). intro. -destruct Hr. destruct H6. unfold g in H4. rewrite H6 in H4. inversion H4. -apply compat_bool_move. - -unfold s. destruct (VertexSet.union_1 H1). -apply VertexSet.union_2. apply VertexSet.diff_3. -assumption. -intro. rewrite Hempty_nmr in H3. elim (VertexSet.empty_1 H3). -apply VertexSet.union_3. assumption. - -(* identically, r is not removed from spillwl, since it is not of high-degree, - but degrees remain the same even if r is not move related *) -set (s := VertexSet.diff (fst (fst (fst wl))) low). -split; intros. -apply (VertexSet.remove_3 H1). -apply VertexSet.remove_2. intro. rewrite <-H2 in H1. clear H2. -unfold s in H1. -generalize (VertexSet.diff_1 H1). intro. -generalize (In_spill_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). intro. -destruct H3. destruct Hr. rewrite H5 in H3. inversion H3. -assumption. - -(* there is no preference edge incident to r *) -split; intros. -rewrite not_incident_edges_1 in H1. destruct H1. assumption. -intros. apply (In_move_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). -rewrite not_incident_edges_1. split. -assumption. -intro. destruct Hr. destruct H4. -cut (move_related g r = true). intro. unfold g in H6. rewrite H4 in H6. inversion H6. -generalize (In_move_props _ _ _ _ _ _ _ _ H1 (refl_equal _) (HWS_irc ircg)). intro. -destruct H6. -apply move_related_charac2 with (e:=a); assumption. -intros. apply (In_move_props _ _ _ _ _ _ _ _ H2 (refl_equal _) (HWS_irc ircg)). - -(* WS_remove_wl respects the invariant *) -assumption. -Qed. - -Lemma WS_simplify : forall r ircg, -VertexSet.In r (get_simplifyWL (irc_wl ircg)) -> -WS_properties (remove_vertex r (irc_g ircg)) (irc_k ircg) - (simplify_wl r ircg (irc_k ircg)). - -Proof. -intros. rewrite <-(Hk ircg). apply WS_simplify_aux. auto. -Qed. diff --git a/backend/Spill_WL.v b/backend/Spill_WL.v deleted file mode 100755 index 1ed1c2d..0000000 --- a/backend/Spill_WL.v +++ /dev/null @@ -1,141 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Freeze_WL. -Require Import Edges. -Require Import WS. -Require Import Interference_adjacency. -Require Import Affinity_relation. -Require Import Remove_Vertex_WL. -Require Import IRC_graph. - -Import RegFacts Props OTFacts. - -Definition spill_wl r ircg K := - let g := irc_g ircg in - let wl := irc_wl ircg in - let simplify := get_simplifyWL wl in - let freeze := get_freezeWL wl in - let spillWL := get_spillWL wl in - let movesWL := get_movesWL wl in - let pre := precolored g in - let int_adj := interference_adj r g in - let not_pre_int_adj := VertexSet.diff int_adj pre in - let pre_adj := preference_adj r g in - let not_pre_pre_adj := VertexSet.diff pre_adj pre in - let newlow := VertexSet.filter (fun x => eq_K K (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj in - let (free, simp) := VertexSet.partition (move_related g) newlow in - let newnmr := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g K x) not_pre_pre_adj in - let simplifyWL__ := VertexSet.union simplify simp in - let simplifyWL' := VertexSet.union simplifyWL__ newnmr in - let freezeWL__ := VertexSet.diff freeze newnmr in - let freezeWL' := VertexSet.union freezeWL__ free in - let spillWL_ := VertexSet.diff spillWL newlow in - let spillWL' := VertexSet.remove r spillWL_ in - let movesWL' := not_incident_edges r movesWL g in - (spillWL', freezeWL', simplifyWL', movesWL'). - - -Lemma WS_spill_aux : forall r ircg, -VertexSet.In r (get_spillWL (irc_wl ircg)) -> -WS_properties (remove_vertex r (irc_g ircg)) (VertexSet.cardinal (pal ircg)) - (spill_wl r ircg (VertexSet.cardinal (pal ircg))). - -Proof. -intros. -generalize (WS_props_equal (remove_vertex r (irc_g ircg)) (VertexSet.cardinal (pal ircg)) - (remove_wl_2 r ircg (VertexSet.cardinal (pal ircg))) - (spill_wl r ircg (VertexSet.cardinal (pal ircg)))). -generalize (WS_remove_wl_2 r ircg). -unfold remove_wl_2, spill_wl, get_simplifyWL, - get_freezeWL, get_spillWL, get_movesWL. -set (g' := remove_vertex r (irc_g ircg)) in *. -set (k := VertexSet.cardinal (pal ircg)) in *. -set (g := irc_g ircg) in *. -set (wl := irc_wl ircg) in *. -set ( simplify := get_simplifyWL wl ) in *. -set ( freeze := get_freezeWL wl ) in *. -set ( spillWL := get_spillWL wl ) in *. -set ( int_adj := interference_adj r g ) in *. -set ( not_pre_int_adj := VertexSet.diff int_adj (precolored g) ) in *. -set ( pre_adj := preference_adj r g ) in *. -set ( not_pre_pre_adj := VertexSet.diff pre_adj (precolored g) ) in *. -set ( low := VertexSet.filter (fun x => eq_K k (VertexSet.cardinal (interference_adj x g))) not_pre_int_adj ) in *. -set ( simpfree := VertexSet.partition (move_related g) low ) in *. -case_eq simpfree. intros free simp Hsf. -unfold simpfree in Hsf. -set ( nmr := VertexSet.filter (fun x => eq_K 1 (VertexSet.cardinal (preference_adj x g)) && has_low_degree g k x) not_pre_pre_adj) in *. -set ( simplifyWL__ := VertexSet.union simplify simp ) in *. -set ( simplifyWL_ := VertexSet.union simplifyWL__ nmr) in *. -set ( simplifyWL' := VertexSet.remove r simplifyWL_ ) in *. -set ( freezeWL__ := VertexSet.diff freeze nmr ) in *. -set ( freezeWL_ := VertexSet.union freezeWL__ free ) in *. -set ( freezeWL' := VertexSet.remove r freezeWL_ ) in *. -set ( spillWL_ := VertexSet.diff spillWL low ) in *. -set ( spillWL' := VertexSet.remove r spillWL_ ) in *. -set ( movesWL' := not_incident_edges r (get_movesWL wl) g) in *. simpl. -generalize (In_spill_props _ _ _ _ _ _ _ _ H (refl_equal _) (HWS_irc ircg)). intro Hr. - -intros HWS H0. apply H0. - -(* r is not removed from simplify *) -split; intros. -apply (VertexSet.remove_3 H1). - -apply VertexSet.remove_2. intro. rewrite <-H2 in H1. clear H2. -destruct (VertexSet.union_1 H1). -destruct (VertexSet.union_1 H2). -generalize (In_simplify_props _ _ _ _ _ _ _ _ H3 (refl_equal _) (HWS_irc ircg)). intro. -destruct H4. destruct Hr. rewrite H4 in H6. inversion H6. - -assert (simp = snd (VertexSet.partition (move_related g) low)) as Hsimp. -rewrite Hsf. auto. rewrite Hsimp in H3. -rewrite VertexSet.partition_2 in H3. -generalize (VertexSet.filter_1 (compat_not_compat (compat_bool_move _)) H3). intro. -unfold low in H4. generalize (VertexSet.filter_1 (eq_K_compat _ _) H4). intro. -elim (not_in_interf_self r g). apply (VertexSet.diff_1 H5). -apply compat_bool_move. - -unfold nmr in H2. generalize (VertexSet.filter_1 (compat_move_up _ _) H2). intro. -elim (not_in_pref_self r g (VertexSet.diff_1 H3)). -assumption. - -(* r is not deleted from freezewl, since it is not move-related and - no vertex is removed from freeze, since free is empty, because - preference adj r g is empty *) - -set (s := VertexSet.union (VertexSet.diff (snd (fst (fst wl))) nmr) free). -split; intros. -apply (VertexSet.remove_3 H1). - -apply VertexSet.remove_2. intro. rewrite <-H2 in H1. clear H2. -unfold s in H1. -destruct (VertexSet.union_1 H1). -generalize (In_freeze_props _ _ _ _ _ _ _ _ (VertexSet.diff_1 H2) (refl_equal _) (HWS_irc ircg)). intro. -destruct H3. destruct Hr. rewrite H5 in H3. inversion H3. - -assert (free = fst (VertexSet.partition (move_related g) low)). -rewrite Hsf. auto. -rewrite H3 in H2. rewrite VertexSet.partition_1 in H2. -generalize (VertexSet.filter_1 (compat_bool_move _ ) H2). intro. -unfold low in H4. generalize (VertexSet.filter_1 (eq_K_compat _ _) H4). intro. -elim (not_in_interf_self r g). apply (VertexSet.diff_1 H5). -apply compat_bool_move. assumption. - -(* spill worklist is unchanged *) -apply VertexSet.eq_refl. - -(* moves worklst is unchanged *) -apply EdgeSet.eq_refl. - -(* WS_remove_wl respects the invariant *) -assumption. -Qed. - -Lemma WS_spill : forall r ircg, -VertexSet.In r (get_spillWL (irc_wl ircg)) -> -WS_properties (remove_vertex r (irc_g ircg)) (irc_k ircg) - (spill_wl r ircg (irc_k ircg)). - -Proof. -intros. rewrite <-(Hk ircg). apply WS_spill_aux. auto. -Qed. diff --git a/backend/Typed_interfgraphs.v b/backend/Typed_interfgraphs.v deleted file mode 100755 index 0f25228..0000000 --- a/backend/Typed_interfgraphs.v +++ /dev/null @@ -1,740 +0,0 @@ -Require Import InterfGraph. -Require Import AST. -Require Import FSets. -Require Import Locations. -Require Import Registers. - -Module Import SRRFacts := Facts SetRegReg. -Module MRegset := FSetAVL.Make OrderedMreg. - -Close Scope nat_scope. - -Definition regregpartition : Type := SetRegReg.t*SetRegReg.t*Regset.t*Regset.t. - -Definition rr1 := fun (p : regregpartition) => fst (fst (fst p)). -Definition rr2 := fun (p : regregpartition) => snd (fst (fst p)). -Definition rr3 := fun (p : regregpartition) => snd (fst p). -Definition rr4 := fun (p : regregpartition) => snd p. - -Definition regreg_edge_type_partition s env := -SetRegReg.fold (fun e s => match env (fst e), env (snd e) with - | Tint, Tint => (SetRegReg.add e (rr1 s), rr2 s, - Regset.add (fst e) (Regset.add (snd e) (rr3 s)), rr4 s) - | Tfloat, Tfloat => (rr1 s, SetRegReg.add e (rr2 s), rr3 s, - Regset.add (fst e) (Regset.add (snd e) (rr4 s))) - | Tint, Tfloat => (rr1 s, rr2 s, Regset.add (fst e) (rr3 s), Regset.add (snd e) (rr4 s)) - | Tfloat, Tint => (rr1 s, rr2 s, Regset.add (snd e) (rr3 s), Regset.add (fst e) (rr4 s)) - end) - s - (SetRegReg.empty, SetRegReg.empty, Regset.empty, Regset.empty). - -Lemma in_partition_in_fst : forall e s env, -SetRegReg.In e (rr1 (regreg_edge_type_partition s env)) -> -SetRegReg.In e s. - -Proof. -intros e s env H. -unfold regreg_edge_type_partition in H. -set (f := fun (e : SetRegReg.elt) (s : regregpartition) => - match env (fst e) with - | Tint => - match env (snd e) with - | Tint => - (SetRegReg.add e (rr1 s), rr2 s, - Regset.add (fst e) (Regset.add (snd e) (rr3 s)), - rr4 s) - | Tfloat => - (rr1 s, rr2 s, Regset.add (fst e) (rr3 s), - Regset.add (snd e) (rr4 s)) - end - | Tfloat => - match env (snd e) with - | Tint => - (rr1 s, rr2 s, Regset.add (snd e) (rr3 s), - Regset.add (fst e) (rr4 s)) - | Tfloat => - (rr1 s, SetRegReg.add e (rr2 s), rr3 s, - Regset.add (fst e) (Regset.add (snd e) (rr4 s))) - end - end). -unfold regregpartition in *. fold f in H. - -assert (forall e set1 set2 set3 set4, SetRegReg.In e (rr1 (SetRegReg.fold f s (set1, set2, set3, set4))) -> - SetRegReg.In e s \/ SetRegReg.In e set1 \/ SetRegReg.In e set2). -clear H. -intros e' s1 s2 s3 s4 H. -rewrite SetRegReg.fold_1 in H. -generalize H. generalize s1 s2 s3 s4. clear s1 s2 s3 s4 H. -generalize (SetRegReg.elements_2). intro HH. -generalize (HH s). clear HH. intro HH. -induction (SetRegReg.elements s). -simpl. right. left. assumption. -intros s1 s2 s3 s4 H. -simpl in H. -assert ((forall x : SetRegReg.elt, - SetoidList.InA (fun x0 y : OrderedRegReg.t => fst x0 = fst y /\ snd x0 = snd y) x l -> - SetRegReg.In x s)). -intros. apply HH. right. assumption. -generalize (IHl H0). clear IHl H0. intro IHl. -assert (f a (s1, s2, s3, s4) = (SetRegReg.add a s1, s2, Regset.add (fst a) (Regset.add (snd a) s3), s4) \/ - f a (s1, s2, s3, s4) = (s1, SetRegReg.add a s2, s3, Regset.add (fst a) (Regset.add (snd a) s4)) \/ - f a (s1, s2, s3, s4) = (s1, s2, Regset.add (fst a) s3, Regset.add (snd a) s4) \/ - f a (s1, s2, s3, s4) = (s1, s2, Regset.add (snd a) s3, Regset.add (fst a) s4)). -unfold f. -destruct (env (snd a)); destruct (env (fst a)); unfold rr1, rr2, rr3, rr4; simpl. -left. reflexivity. -right. right. right. reflexivity. -right. right. left. reflexivity. -right. left. reflexivity. -destruct H0. -rewrite H0 in H. - -generalize (IHl (SetRegReg.add a s1) s2 _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. - -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. left. assumption. -right. right. assumption. -destruct H0. -rewrite H0 in H. -generalize (IHl s1 (SetRegReg.add a s2) _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. right. assumption. -destruct H0. -rewrite H0 in H. -generalize (IHl s1 s2 _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -right. right. assumption. -rewrite H0 in H. -generalize (IHl s1 s2 _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -right. right. assumption. -generalize (H0 _ _ _ _ _ H). clear H0. intro H0. -destruct H0. -assumption. -destruct H0; elim (SetRegReg.empty_1 H0). -Qed. - -Lemma in_partition_in_snd : forall e s env, -SetRegReg.In e (rr2 (regreg_edge_type_partition s env)) -> -SetRegReg.In e s. - -Proof. -intros e s env H. -unfold regreg_edge_type_partition in H. -set (f := fun (e : SetRegReg.elt) (s : regregpartition) => - match env (fst e) with - | Tint => - match env (snd e) with - | Tint => - (SetRegReg.add e (rr1 s), rr2 s, - Regset.add (fst e) (Regset.add (snd e) (rr3 s)), - rr4 s) - | Tfloat => - (rr1 s, rr2 s, Regset.add (fst e) (rr3 s), - Regset.add (snd e) (rr4 s)) - end - | Tfloat => - match env (snd e) with - | Tint => - (rr1 s, rr2 s, Regset.add (snd e) (rr3 s), - Regset.add (fst e) (rr4 s)) - | Tfloat => - (rr1 s, SetRegReg.add e (rr2 s), rr3 s, - Regset.add (fst e) (Regset.add (snd e) (rr4 s))) - end - end). -unfold regregpartition in *. fold f in H. - -assert (forall e set1 set2 set3 set4, SetRegReg.In e (rr2 (SetRegReg.fold f s (set1, set2, set3, set4))) -> - SetRegReg.In e s \/ SetRegReg.In e set1 \/ SetRegReg.In e set2). -clear H. -intros e' s1 s2 s3 s4 H. -rewrite SetRegReg.fold_1 in H. -generalize H. generalize s1 s2 s3 s4. clear s1 s2 s3 s4 H. -generalize (SetRegReg.elements_2). intro HH. -generalize (HH s). clear HH. intro HH. -induction (SetRegReg.elements s). -simpl. right. right. assumption. -intros s1 s2 s3 s4 H. -simpl in H. -assert ((forall x : SetRegReg.elt, - SetoidList.InA (fun x0 y : OrderedRegReg.t => fst x0 = fst y /\ snd x0 = snd y) x l -> - SetRegReg.In x s)). -intros. apply HH. right. assumption. -generalize (IHl H0). clear IHl H0. intro IHl. -assert (f a (s1, s2, s3, s4) = (SetRegReg.add a s1, s2, Regset.add (fst a) (Regset.add (snd a) s3), s4) \/ - f a (s1, s2, s3, s4) = (s1, SetRegReg.add a s2, s3, Regset.add (fst a) (Regset.add (snd a) s4)) \/ - f a (s1, s2, s3, s4) = (s1, s2, Regset.add (fst a) s3, Regset.add (snd a) s4) \/ - f a (s1, s2, s3, s4) = (s1, s2, Regset.add (snd a) s3, Regset.add (fst a) s4)). -unfold f. -destruct (env (snd a)); destruct (env (fst a)); unfold rr1, rr2, rr3, rr4; simpl. -left. reflexivity. -right. right. right. reflexivity. -right. right. left. reflexivity. -right. left. reflexivity. -destruct H0. -rewrite H0 in H. - -generalize (IHl (SetRegReg.add a s1) s2 _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. - -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. left. assumption. -right. right. assumption. -destruct H0. -rewrite H0 in H. -generalize (IHl s1 (SetRegReg.add a s2) _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. right. assumption. -destruct H0. -rewrite H0 in H. -generalize (IHl s1 s2 _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -right. right. assumption. -rewrite H0 in H. -generalize (IHl s1 s2 _ _ H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -right. right. assumption. -generalize (H0 _ _ _ _ _ H). clear H0. intro H0. -destruct H0. -assumption. -destruct H0; elim (SetRegReg.empty_1 H0). -Qed. - -Lemma in_partition_type_fst : forall e s env, -SetRegReg.In e (rr1 (regreg_edge_type_partition s env)) -> -env (fst e) = Tint /\ env (snd e) = Tint. - -Proof. -intros e s env H. -unfold regreg_edge_type_partition in H. -set (f := fun (e : SetRegReg.elt) (s : regregpartition) => - match env (fst e) with - | Tint => - match env (snd e) with - | Tint => - (SetRegReg.add e (rr1 s), rr2 s, - Regset.add (fst e) (Regset.add (snd e) (rr3 s)), - rr4 s) - | Tfloat => - (rr1 s, rr2 s, Regset.add (fst e) (rr3 s), - Regset.add (snd e) (rr4 s)) - end - | Tfloat => - match env (snd e) with - | Tint => - (rr1 s, rr2 s, Regset.add (snd e) (rr3 s), - Regset.add (fst e) (rr4 s)) - | Tfloat => - (rr1 s, SetRegReg.add e (rr2 s), rr3 s, - Regset.add (fst e) (Regset.add (snd e) (rr4 s))) - end - end). -unfold regregpartition in *. fold f in H. - -cut (forall e set1 set2 set3 set4, SetRegReg.In e (rr1 (SetRegReg.fold f s (set1, set2, set3, set4))) -> - (env (fst e) = Tint /\ env (snd e) = Tint) \/ SetRegReg.In e set1). - -intros. -generalize (H0 _ _ _ _ _ H). clear H H0. intro H. -destruct H. assumption. elim (SetRegReg.empty_1 H). - -(* cut property *) -intros. -generalize H0. clear H H0. intro H. -rewrite SetRegReg.fold_1 in H. -generalize H. clear H. generalize set1 set2 set3 set4. -induction (SetRegReg.elements s); intros s1 s2 s3 s4 H. -simpl in H. right. assumption. -simpl in H. - -assert ((f a (s1, s2, s3, s4) = (SetRegReg.add a s1, s2, Regset.add (fst a) (Regset.add (snd a) s3), s4) /\ - env (fst a) = Tint /\ - env (snd a) = Tint)\/ - (f a (s1, s2, s3, s4) = (s1, SetRegReg.add a s2, s3, Regset.add (fst a) (Regset.add (snd a) s4)) /\ - env (fst a) = Tfloat /\ - env (snd a) = Tfloat) \/ - f a (s1, s2, s3, s4) = (s1,s2, Regset.add (fst a) s3, Regset.add (snd a) s4) \/ - f a (s1, s2, s3, s4) = (s1,s2, Regset.add (snd a) s3, Regset.add (fst a) s4) -). -unfold f. -destruct (env (snd a)); destruct (env (fst a)); unfold rr1, rr2, rr3, rr4; simpl. -left. auto. -right. right. right. reflexivity. -right. right. left. reflexivity. -right. left. auto. - -destruct H0. -destruct H0. destruct H1. rewrite H0 in H. -generalize (IHl (SetRegReg.add a s1) s2 _ _ H). intro H3. -destruct H3. -left. assumption. -destruct (proj1 (add_iff _ _ _) H3). -left. intuition. rewrite <-H5. auto. rewrite <-H6. auto. -right. assumption. - -destruct H0. destruct H0. destruct H1. rewrite H0 in H. -apply (IHl s1 (SetRegReg.add a s2) _ _ H). - -destruct H0. rewrite H0 in H. apply (IHl s1 s2 _ _ H). -rewrite H0 in H. apply (IHl s1 s2 _ _ H). -Qed. - -Lemma in_partition_type_snd : forall e s env, -SetRegReg.In e (rr2 (regreg_edge_type_partition s env)) -> -env (fst e) = Tfloat /\ env (snd e) = Tfloat. - -Proof. -intros e s env H. -unfold regreg_edge_type_partition in H. -set (f := fun (e : SetRegReg.elt) (s : regregpartition) => - match env (fst e) with - | Tint => - match env (snd e) with - | Tint => - (SetRegReg.add e (rr1 s), rr2 s, - Regset.add (fst e) (Regset.add (snd e) (rr3 s)), - rr4 s) - | Tfloat => - (rr1 s, rr2 s, Regset.add (fst e) (rr3 s), - Regset.add (snd e) (rr4 s)) - end - | Tfloat => - match env (snd e) with - | Tint => - (rr1 s, rr2 s, Regset.add (snd e) (rr3 s), - Regset.add (fst e) (rr4 s)) - | Tfloat => - (rr1 s, SetRegReg.add e (rr2 s), rr3 s, - Regset.add (fst e) (Regset.add (snd e) (rr4 s))) - end - end). -unfold regregpartition in *. fold f in H. - -cut (forall e set1 set2 set3 set4, SetRegReg.In e (rr2 (SetRegReg.fold f s (set1, set2, set3, set4))) -> - (env (fst e) = Tfloat /\ env (snd e) = Tfloat) \/ SetRegReg.In e set2). - -intros. -generalize (H0 _ _ _ _ _ H). clear H H0. intro H. -destruct H. assumption. elim (SetRegReg.empty_1 H). - -(* cut property *) -intros. -generalize H0. clear H H0. intro H. -rewrite SetRegReg.fold_1 in H. -generalize H. clear H. generalize set1 set2 set3 set4. -induction (SetRegReg.elements s); intros s1 s2 s3 s4 H. -simpl in H. right. assumption. -simpl in H. - -assert ((f a (s1, s2, s3, s4) = (SetRegReg.add a s1, s2, Regset.add (fst a) (Regset.add (snd a) s3), s4) /\ - env (fst a) = Tint /\ - env (snd a) = Tint)\/ - (f a (s1, s2, s3, s4) = (s1, SetRegReg.add a s2, s3, Regset.add (fst a) (Regset.add (snd a) s4)) /\ - env (fst a) = Tfloat /\ - env (snd a) = Tfloat) \/ - f a (s1, s2, s3, s4) = (s1,s2, Regset.add (fst a) s3, Regset.add (snd a) s4) \/ - f a (s1, s2, s3, s4) = (s1,s2, Regset.add (snd a) s3, Regset.add (fst a) s4) -). -unfold f. -destruct (env (snd a)); destruct (env (fst a)); unfold rr1, rr2, rr3, rr4; simpl. -left. auto. -right. right. right. reflexivity. -right. right. left. reflexivity. -right. left. auto. - -destruct H0. -destruct H0. rewrite H0 in H. -apply (IHl (SetRegReg.add a s1) s2 _ _ H). -destruct H0. destruct H0. rewrite H0 in H. -generalize (IHl s1 (SetRegReg.add a s2) _ _ H). intro. -destruct H2. -left. assumption. -destruct (proj1 (add_iff _ _ _) H2). -left. intuition. rewrite <-H1. auto. rewrite <-H6. auto. -right. assumption. - -destruct H0. rewrite H0 in H. apply (IHl _ _ _ _ H). - -rewrite H0 in H. apply (IHl s1 s2 _ _ H). -Qed. - -Definition regmregpartition : Type := SetRegMreg.t*SetRegMreg.t*Regset.t*Regset.t*MRegset.t*MRegset.t. - -Definition rm1 := fun (p : regmregpartition) => fst (fst (fst (fst (fst p)))). -Definition rm2 := fun (p : regmregpartition) => snd (fst (fst (fst (fst p)))). -Definition rm3 := fun (p : regmregpartition) => snd (fst (fst (fst p))). -Definition rm4 := fun (p : regmregpartition) => snd (fst (fst p)). -Definition rm5 := fun (p : regmregpartition) => snd (fst p). -Definition rm6 := fun (p : regmregpartition) => snd p. - -Module Import SRMFacts := Facts SetRegMreg. - -Definition regmreg_edge_type_partition s env := -SetRegMreg.fold (fun e s => match env (fst e), mreg_type (snd e) with - | Tint, Tint => (SetRegMreg.add e (rm1 s), rm2 s, - Regset.add (fst e) (rm3 s), rm4 s, - MRegset.add (snd e) (rm5 s), rm6 s) - | Tfloat, Tfloat => (rm1 s, SetRegMreg.add e (rm2 s), - rm3 s, Regset.add (fst e) (rm4 s), - rm5 s, MRegset.add (snd e) (rm6 s)) - | Tint, Tfloat => (rm1 s, rm2 s, - Regset.add (fst e) (rm3 s), rm4 s, - rm5 s, MRegset.add (snd e) (rm6 s)) - |Tfloat, Tint => (rm1 s, rm2 s, - rm3 s, Regset.add (fst e) (rm4 s), - MRegset.add (snd e) (rm5 s), rm6 s) - end) -s -(SetRegMreg.empty, SetRegMreg.empty, Regset.empty, Regset.empty, MRegset.empty, MRegset.empty). - -Lemma in_mreg_partition_in_fst : forall e s env, -SetRegMreg.In e (rm1 (regmreg_edge_type_partition s env)) -> -SetRegMreg.In e s. - -Proof. -Admitted. -(* -intros e s env H. -unfold regmreg_edge_type_partition in H. -set (f := (fun (e : SetRegMreg.elt) (s : SetRegMreg.t * SetRegMreg.t) => - match env (fst e) with - | Tint => - match mreg_type (snd e) with - | Tint => (SetRegMreg.add e (fst s), snd s) - | Tfloat => s - end - | Tfloat => - match mreg_type (snd e) with - | Tint => s - | Tfloat => (fst s, SetRegMreg.add e (snd s)) - end - end)) in H. - -assert (forall e set1 set2, SetRegMreg.In e (fst (SetRegMreg.fold f s (set1, set2))) -> - SetRegMreg.In e s \/ SetRegMreg.In e set1 \/ SetRegMreg.In e set2). -clear H. -intros e' s1 s2 H. -rewrite SetRegMreg.fold_1 in H. -generalize H. generalize s1 s2. clear s1 s2 H. -generalize (SetRegMreg.elements_2). intro HH. -generalize (HH s). clear HH. intro HH. -induction (SetRegMreg.elements s). -simpl. right. left. assumption. -intros s1 s2 H. -simpl in H. -assert ((forall x : SetRegMreg.elt, - SetoidList.InA (fun x0 y : OrderedRegMreg.t => fst x0 = fst y /\ snd x0 = snd y) x l -> - SetRegMreg.In x s)). -intros. apply HH. right. assumption. -generalize (IHl H0). clear IHl H0. intro IHl. -assert (f a (s1, s2) = (SetRegMreg.add a s1, s2) \/ - f a (s1, s2) = (s1, SetRegMreg.add a s2) \/ - f a (s1, s2) = (s1,s2)). -unfold f. -destruct (mreg_type (snd a)); destruct (env (fst a)); simpl. -left. reflexivity. -right. right. reflexivity. -right. right. reflexivity. -right. left. reflexivity. -destruct H0. -rewrite H0 in H. - -generalize (IHl (SetRegMreg.add a s1) s2 H). -intro H1. destruct H1. -left. assumption. -destruct H1. - -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. left. assumption. -right. right. assumption. -destruct H0. -rewrite H0 in H. -generalize (IHl s1 (SetRegMreg.add a s2) H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. right. assumption. -rewrite H0 in H. -generalize (IHl s1 s2 H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -right. right. assumption. -generalize (H0 _ _ _ H). clear H0. intro H0. -destruct H0. -assumption. -destruct H0; elim (SetRegMreg.empty_1 H0). -Qed. -*) - -Lemma in_mreg_partition_in_snd : forall e s env, -SetRegMreg.In e (rm2 (regmreg_edge_type_partition s env)) -> -SetRegMreg.In e s. - -Proof. -Admitted. -(* -intros e s env H. -unfold regmreg_edge_type_partition in H. -set (f := (fun (e : SetRegMreg.elt) (s : SetRegMreg.t * SetRegMreg.t) => - match env (fst e) with - | Tint => - match mreg_type (snd e) with - | Tint => (SetRegMreg.add e (fst s), snd s) - | Tfloat => s - end - | Tfloat => - match mreg_type (snd e) with - | Tint => s - | Tfloat => (fst s, SetRegMreg.add e (snd s)) - end - end)) in H. - -assert (forall e set1 set2, SetRegMreg.In e (snd (SetRegMreg.fold f s (set1, set2))) -> - SetRegMreg.In e s \/ SetRegMreg.In e set1 \/ SetRegMreg.In e set2). -clear H. -intros e' s1 s2 H. -rewrite SetRegMreg.fold_1 in H. -generalize H. generalize s1 s2. clear s1 s2 H. -generalize (SetRegMreg.elements_2). intro HH. -generalize (HH s). clear HH. intro HH. -induction (SetRegMreg.elements s). -simpl. right. right. assumption. -intros s1 s2 H. -simpl in H. -assert ((forall x : SetRegMreg.elt, - SetoidList.InA (fun x0 y : OrderedRegMreg.t => fst x0 = fst y /\ snd x0 = snd y) x l -> - SetRegMreg.In x s)). -intros. apply HH. right. assumption. -generalize (IHl H0). clear IHl H0. intro IHl. -assert (f a (s1, s2) = (SetRegMreg.add a s1, s2) \/ - f a (s1, s2) = (s1, SetRegMreg.add a s2) \/ - f a (s1, s2) = (s1,s2)). -unfold f. -destruct (mreg_type (snd a)); destruct (env (fst a)); simpl. -left. reflexivity. -right. right. reflexivity. -right. right. reflexivity. -right. left. reflexivity. -destruct H0. -rewrite H0 in H. - -generalize (IHl (SetRegMreg.add a s1) s2 H). -intro H1. destruct H1. -left. assumption. -destruct H1. - -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. left. assumption. -right. right. assumption. -destruct H0. -rewrite H0 in H. -generalize (IHl s1 (SetRegMreg.add a s2) H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -destruct (proj1 (add_iff _ _ _) H1). -left. apply HH. left. intuition. -right. right. assumption. -rewrite H0 in H. -generalize (IHl s1 s2 H). -intro H1. destruct H1. -left. assumption. -destruct H1. -right. left. assumption. -right. right. assumption. -generalize (H0 _ _ _ H). clear H0. intro H0. -destruct H0. -assumption. -destruct H0; elim (SetRegMreg.empty_1 H0). -Qed. -*) - -Lemma in_mreg_partition_type_fst : forall e s env, -SetRegMreg.In e (rm1 (regmreg_edge_type_partition s env)) -> -env (fst e) = Tint /\ mreg_type (snd e) = Tint. - -Proof. -Admitted. -(* -intros e s env H. -unfold regmreg_edge_type_partition in H. -set (f := (fun (e : SetRegMreg.elt) (s : SetRegMreg.t * SetRegMreg.t) => - match env (fst e) with - | Tint => - match mreg_type (snd e) with - | Tint => (SetRegMreg.add e (fst s), snd s) - | Tfloat => s - end - | Tfloat => - match mreg_type (snd e) with - | Tint => s - | Tfloat => (fst s, SetRegMreg.add e (snd s)) - end - end)) in H. - -cut (forall e set1 set2, SetRegMreg.In e (fst (SetRegMreg.fold f s (set1, set2))) -> - (env (fst e) = Tint /\ mreg_type (snd e) = Tint) \/ SetRegMreg.In e set1). - -intros. -generalize (H0 _ _ _ H). clear H H0. intro H. -destruct H. assumption. elim (SetRegMreg.empty_1 H). - -(* cut property *) -intros. -generalize H0. clear H H0. intro H. -rewrite SetRegMreg.fold_1 in H. -generalize H. clear H. generalize set1 set2. -induction (SetRegMreg.elements s); intros s1 s2 H. -simpl in H. right. assumption. -simpl in H. - -assert ((f a (s1, s2) = (SetRegMreg.add a s1, s2) /\ - env (fst a) = Tint /\ - mreg_type (snd a) = Tint)\/ - (f a (s1, s2) = (s1, SetRegMreg.add a s2) /\ - env (fst a) = Tfloat /\ - mreg_type (snd a) = Tfloat) \/ - f a (s1, s2) = (s1,s2)). -unfold f. -destruct (mreg_type (snd a)); destruct (env (fst a)); simpl. -left. auto. -right. right. reflexivity. -right. right. reflexivity. -right. left. auto. - -destruct H0. -destruct H0. destruct H1. rewrite H0 in H. -generalize (IHl (SetRegMreg.add a s1) s2 H). intro H3. -destruct H3. -left. assumption. -destruct (proj1 (add_iff _ _ _) H3). -left. intuition. rewrite <-H5. auto. rewrite <-H6. auto. -right. assumption. - -destruct H0. destruct H0. destruct H1. rewrite H0 in H. -apply (IHl s1 (SetRegMreg.add a s2) H). - -rewrite H0 in H. -apply (IHl s1 s2 H). -Qed. -*) - -Lemma in_mreg_partition_type_snd : forall e s env, -SetRegMreg.In e (rm2 (regmreg_edge_type_partition s env)) -> -env (fst e) = Tfloat /\ mreg_type (snd e) = Tfloat. - -Proof. -Admitted. -(* -intros e s env H. -unfold regmreg_edge_type_partition in H. -set (f := (fun (e : SetRegMreg.elt) (s : SetRegMreg.t * SetRegMreg.t) => - match env (fst e) with - | Tint => - match mreg_type (snd e) with - | Tint => (SetRegMreg.add e (fst s), snd s) - | Tfloat => s - end - | Tfloat => - match mreg_type (snd e) with - | Tint => s - | Tfloat => (fst s, SetRegMreg.add e (snd s)) - end - end)) in H. - -cut (forall e set1 set2, SetRegMreg.In e (snd (SetRegMreg.fold f s (set1, set2))) -> - (env (fst e) = Tfloat /\ mreg_type (snd e) = Tfloat) \/ SetRegMreg.In e set2). - -intros. -generalize (H0 _ _ _ H). clear H H0. intro H. -destruct H. assumption. elim (SetRegMreg.empty_1 H). - -(* cut property *) -intros. -generalize H0. clear H H0. intro H. -rewrite SetRegMreg.fold_1 in H. -generalize H. clear H. generalize set1 set2. -induction (SetRegMreg.elements s); intros s1 s2 H. -simpl in H. right. assumption. -simpl in H. - -assert ((f a (s1, s2) = (SetRegMreg.add a s1, s2) /\ - env (fst a) = Tint /\ - mreg_type (snd a) = Tint)\/ - (f a (s1, s2) = (s1, SetRegMreg.add a s2) /\ - env (fst a) = Tfloat /\ - mreg_type (snd a) = Tfloat) \/ - f a (s1, s2) = (s1,s2)). -unfold f. -destruct (mreg_type (snd a)); destruct (env (fst a)); simpl. -left. auto. -right. right. reflexivity. -right. right. reflexivity. -right. left. auto. - -destruct H0. -destruct H0. destruct H1. rewrite H0 in H. -apply (IHl (SetRegMreg.add a s1) s2 H). - -destruct H0. destruct H0. destruct H1. rewrite H0 in H. -generalize (IHl s1 (SetRegMreg.add a s2) H). intro. -destruct H3. -left. auto. -destruct (proj1 (add_iff _ _ _) H3). -destruct H4. left. rewrite H4 in *. rewrite H5 in *. intuition. -right. auto. -rewrite H0 in H. apply (IHl s1 s2 H). -Qed. - -Definition Typed_interfgraphs g env := -let (int_regreg_interf, float_regreg_interf) := - regreg_edge_type_partition (interf_reg_reg g) env in -let (int_regmreg_interf, float_regmreg_interf) := - regmreg_edge_type_partition (interf_reg_mreg g) env in -let (int_regreg_pref, float_regreg_pref) := - regreg_edge_type_partition (pref_reg_reg g) env in -let (int_regmreg_pref, float_regmreg_pref) := - regmreg_edge_type_partition (pref_reg_mreg g) env in -(mkgraph int_regreg_interf int_regmreg_interf int_regreg_pref int_regmreg_pref, - mkgraph float_regreg_interf float_regmreg_interf float_regreg_pref float_regmreg_pref). -*) - - diff --git a/backend/WS.v b/backend/WS.v deleted file mode 100755 index 2f43327..0000000 --- a/backend/WS.v +++ /dev/null @@ -1,231 +0,0 @@ -Require Import FSets. -Require Import InterfGraphMapImp. -Require Import Conservative_criteria. -Require Import Edges. -Require Import MyRegisters. -Require Import Affinity_relation. - -Import Edge RegFacts Props OTFacts. - -(* Intersections of vertices sets of the worklists are empty *) -Lemma WS_empty_inter_1 : forall g palette WL, -WS_properties g palette WL -> -VertexSet.Empty (VertexSet.inter (get_spillWL WL) (get_freezeWL WL)). - -Proof. -intros g palette WL H. -unfold VertexSet.Empty. -intros a H0. -generalize (VertexSet.inter_1 H0);intro H1. -generalize (VertexSet.inter_2 H0);intro H2. -unfold WS_properties in H. -destruct H as [H H3];destruct H3 as [H3 H4];destruct H4 as [H4 H5]. -generalize (proj1 (H a) H1);intro H6. -destruct H6 as [H6 _]. -generalize (proj1 (H3 a) H2);intro H7. -destruct H7 as [H7 _]. -rewrite H6 in H7; inversion H7. -Qed. - -Lemma WS_empty_inter_2 : forall g palette WL, -WS_properties g palette WL -> -VertexSet.Empty (VertexSet.inter (get_spillWL WL) (get_simplifyWL WL)). - -Proof. -intros g palette WL H. -unfold VertexSet.Empty. -intros a H0. -generalize (VertexSet.inter_1 H0);intro H1. -generalize (VertexSet.inter_2 H0);intro H2. -unfold WS_properties in H. -destruct H as [H H3];destruct H3 as [H3 H4];destruct H4 as [H4 H5]. -generalize (proj1 (H a) H1);intro H6. -destruct H6 as [H6 _]. -generalize (proj1 (H4 a) H2);intro H7. -destruct H7 as [H7 _]. -rewrite H6 in H7; inversion H7. -Qed. - -Lemma WS_empty_inter_3 : forall g palette WL, -WS_properties g palette WL -> -VertexSet.Empty (VertexSet.inter (get_freezeWL WL) (get_simplifyWL WL)). - -Proof. -intros g palette WL H. -unfold VertexSet.Empty. -intros a H0. -generalize (VertexSet.inter_1 H0);intro H1. -generalize (VertexSet.inter_2 H0);intro H2. -unfold WS_properties in H. -destruct H as [H H3];destruct H3 as [H3 H4];destruct H4 as [H4 H5]. -generalize (proj1 (H3 a) H1);intro H6. -destruct H6 as [_ H6]. -destruct H6 as [H6 _]. -generalize (proj1 (H4 a) H2);intro H7. -destruct H7 as [_ H7]. -destruct H7 as [H7 _]. -rewrite H6 in H7; inversion H7. -Qed. - -(* A tactic for simplifying proofs of belonging of a vertex to a worklist *) -Ltac WS_apply H := generalize H;intro HWS_; -match goal with -| |- (VertexSet.In ?A _) => destruct HWS_ as [HWS_ HWS__]; - try (apply (proj2 (HWS_ A))); - destruct HWS__ as [HWS__ HWS___]; - try (apply (proj2 (HWS__ A))); - destruct HWS___ as [HWS___ HWS____]; - try (apply (proj2 (HWS___ A))); - clear HWS_ HWS__ HWS___ HWS____ -| |- (EdgeSet.In ?A _) => do 3 destruct HWS_ as [_ HWS_]; - apply (proj2 (HWS_ A)); - clear HWS_ -end. - -(* Lemmas for generalizing properties of a vertex belonging to a given worklist *) -Lemma In_spill_props : forall x g WL s a b c palette, -VertexSet.In x s -> -WL = (s,a,b,c) -> -WS_properties g palette WL -> -has_low_degree g palette x = false /\ In_graph x g /\ ~VertexSet.In x (precolored g). - -Proof. -intros x g WL s a b c palette H H0 H1. -unfold WS_properties in H1;rewrite H0 in H1. -destruct H1 as [H1 _]. -unfold get_spillWL in H1;simpl in H1. -apply (proj1 (H1 x) H). -Qed. - -Lemma In_freeze_props : forall x g WL s a b c palette, -VertexSet.In x s -> -WL = (a,s,b,c) -> -WS_properties g palette WL -> -has_low_degree g palette x = true /\ move_related g x = true /\ In_graph x g /\ ~VertexSet.In x (precolored g). - -Proof. -intros x g WL s a b c palette H H0 H1. -unfold WS_properties in H1;rewrite H0 in H1. -destruct H1 as [_ H1]. -destruct H1 as [H1 _]. -unfold get_freezeWL in H1;simpl in H1. -generalize (proj1 (H1 x) H);intro. -intuition. -apply move_related_in_graph;intuition. -Qed. - -Lemma In_simplify_props : forall x g WL s a b c palette, -VertexSet.In x s -> -WL = (a,b,s,c) -> -WS_properties g palette WL -> -has_low_degree g palette x = true /\ move_related g x = false /\ In_graph x g /\ ~VertexSet.In x (precolored g). - -Proof. -intros x g WL s a b c palette H H0 H1. -unfold WS_properties in H1;rewrite H0 in H1. -do 2 destruct H1 as [_ H1]. -destruct H1 as [H1 _]. -unfold get_spillWL in H1;simpl in H1. -apply (proj1 (H1 x) H). -Qed. - -Lemma In_move_props : forall e g WL s a b c palette, -EdgeSet.In e s -> -WL = (a,b,c,s) -> -WS_properties g palette WL -> -aff_edge e /\ In_graph_edge e g. - -Proof. -intros e g WL s a b c palette H H0 H1. -unfold WS_properties in H1;rewrite H0 in H1. -do 3 destruct H1 as [_ H1]. -unfold get_movesWL in H1;simpl in H1. -apply (proj1 (H1 e) H). -Qed. - -Lemma WS_props_equal : -forall g palette ws ws', -VertexSet.Equal (get_simplifyWL ws) (get_simplifyWL ws') -> -VertexSet.Equal (get_freezeWL ws) (get_freezeWL ws') -> -VertexSet.Equal (get_spillWL ws) (get_spillWL ws') -> -EdgeSet.Equal (get_movesWL ws) (get_movesWL ws') -> -WS_properties g palette ws -> -WS_properties g palette ws'. - -Proof. -unfold WS_properties;unfold get_spillWL;unfold get_freezeWL; -unfold get_simplifyWL;unfold get_movesWL;simpl;unfold VertexSet.Equal; -unfold EdgeSet.Equal;intros g palette ws ws' H H0 H1 H2 H3. -destruct ws as [ws d]; destruct ws as [ws c]; destruct ws as [a b]. -destruct ws' as [ws' p]; destruct ws' as [ws' o]; destruct ws' as [m n]. simpl in *. -generalize (VertexSet.eq_sym H);generalize (VertexSet.eq_sym H0); -generalize (VertexSet.eq_sym H1);generalize (EdgeSet.eq_sym H2); -clear H;clear H0;clear H1;clear H2;intros H2 H1 H0 H. - -destruct H3 as [Hsp H3];destruct H3 as [Hf H3]; -destruct H3 as [Hsi Hm]. -do 2 split. -intro H4;apply (proj1 (Hsp x) (proj1 (H1 x) H4)). -intro H4;apply (proj2 (H1 x) (proj2 (Hsp x) H4)). -split;intro H4. -apply (proj1 (Hf x) (proj1 (H0 x) H4)). -apply (proj2 (H0 x) (proj2 (Hf x) H4)). -split. -split;intro H4. -apply (proj1 (Hsi x) (proj1 (H x) H4)). -apply (proj2 (H x) (proj2 (Hsi x) H4)). -split;intro H4. -apply (proj1 (Hm e) (proj1 (H2 e) H4)). -apply (proj2 (H2 e) (proj2 (Hm e) H4)). -Qed. - -(* Definition of the nonprecolored vertices of a graph *) -Definition not_precolored g := VertexSet.diff (V g) (precolored g). - -(* The union of vertices worklists forms the nonprecolored vertices set of g *) -Lemma not_precolored_union_ws : forall g palette ws, -WS_properties g palette ws -> -VertexSet.Equal -(VertexSet.union (VertexSet.union (get_spillWL ws) (get_freezeWL ws)) (get_simplifyWL ws)) -(not_precolored g). - -Proof. -intros g palette ws HWS. -split. intro H. -unfold not_precolored. apply VertexSet.diff_3. -destruct (VertexSet.union_1 H). -destruct (VertexSet.union_1 H0). -apply (proj1(proj2 (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). -apply (proj2(proj2 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). -apply (proj2(proj2 (In_simplify_props _ _ _ _ _ _ _ _ H0 (refl_equal _) HWS))). -destruct (VertexSet.union_1 H). -destruct (VertexSet.union_1 H0). -apply (proj2(proj2 (In_spill_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). -apply (proj2(proj2 (In_freeze_props _ _ _ _ _ _ _ _ H1 (refl_equal _) HWS))). -apply (proj2(proj2 (In_simplify_props _ _ _ _ _ _ _ _ H0 (refl_equal _) HWS))). -intro H. -unfold not_precolored in H. -generalize (VertexSet.diff_1 H). intro H0. -generalize (VertexSet.diff_2 H). clear H. intro H. -case_eq (has_low_degree g palette a); intro Hlow. -case_eq (move_related g a); intro Haff. -apply VertexSet.union_2. apply VertexSet.union_3. -WS_apply HWS. intuition. -apply VertexSet.union_3. -WS_apply HWS. intuition. -apply VertexSet.union_2. apply VertexSet.union_2. -WS_apply HWS. intuition. -Qed. - -(* The moves worklists is equal to the set of affinity edges *) -Lemma moves_AE : forall g palette ws, -WS_properties g palette ws -> -EdgeSet.Equal (AE g) (get_movesWL ws). - -Proof. -split; intros. -destruct ws. destruct p. destruct p. -simpl. WS_apply H. apply (proj1 (In_graph_aff_edge_in_AE _ _) H0). -generalize (In_move_props _ _ _ _ _ _ _ _ H0 (refl_equal _) H). intro H1. -apply (proj2 (In_graph_aff_edge_in_AE _ _) H1). -Qed. |